Relational Algebra and Relational Calculus - cs.duke.edu · Recap: SQL --Lecture 2/3 •Creating/modifying relations •Specifying integrity constraints •Key/candidate key, superkey,

CompSci 516DatabaseSystems

Lecture4RelationalAlgebra

andRelationalCalculus

Instructor:Sudeepa Roy

1DukeCS,Fall2018 CompSci516:DatabaseSystems

Announcements• Reminder:HW1

– Sakai:Resources->HW->HW1folder– Dueon09/20(Thurs),11:55pm,nolatedays– Startnow!– Submissioninstructionsforgradescope tobeupdated(willbenotifiedthroughpiazza)

• Yourpiazzaandsakai accountsshouldbeactive– Lastcall!willusepiazzaforpop-upquizzes– ifnotonpiazza,sendmeanemail– Installpiazzaapponyourphone(orbringalaptop)

DukeCS,Fall2018 CompSci516:DatabaseSystems 2

Recap:SQL-- Lecture2/3

• Creating/modifyingrelations• Specifyingintegrityconstraints• Key/candidatekey,superkey,primarykey,foreignkey• ConceptualevaluationofSQLqueries• Joins• Groupbys andaggregates• Nestedqueries• NULLs• Views


Onwhiteboard:Fromlastlecture

Correct/incorrectgroup-byqueries

Today’stopics

• RelationalAlgebra(RA)andRelationalCalculus(RC)

• Readingmaterial– [RG]Chapter4(RA,RC)– [GUW]Chapters2.4,5.1,5.2


Acknowledgement:Thefollowingslideshavebeencreatedadaptingtheinstructormaterialofthe[RG]bookprovidedbytheauthorsDr.Ramakrishnan andDr.Gehrke.

RelationalQueryLanguages


RelationalQueryLanguages

• Querylanguages:Allowmanipulationandretrievalofdatafromadatabase

• Relationalmodelsupportssimple,powerfulQLs:– Strongformalfoundationbasedonlogic– Allowsformuchoptimization

• QueryLanguages!= programminglanguages– QLsnotintendedtobeusedforcomplexcalculations– QLssupporteasy,efficientaccesstolargedatasets


FormalRelationalQueryLanguages

• Two“mathematical”QueryLanguagesformthebasisfor“real”languages(e.g.SQL),andforimplementation:– RelationalAlgebra:Moreoperational,veryusefulforrepresentingexecutionplans

– RelationalCalculus:Letsusersdescribewhattheywant,ratherthanhowtocomputeit(Non-operational,declarative,orprocedural)

• Note:Declarative(RC,SQL)vs.Operational(RA)


Preliminaries(recap)• Aqueryisappliedtorelationinstances,andtheresultofaqueryisalsoarelationinstance.– Schemasofinputrelationsforaqueryarefixed

• querywillrunregardlessofinstance

– Theschemafortheresultofagivenqueryisalsofixed• Determinedbydefinitionofquerylanguageconstructs

• Positionalvs.named-fieldnotation:– Positionalnotationeasierforformaldefinitions,named-fieldnotationmorereadable


ExampleSchemaandInstances

sid sname rating age22 dustin 7 45.031 lubber 8 55.558 rusty 10 35.0

sid sname rating age28 yuppy 9 35.031 lubber 8 55.544 guppy 5 35.058 rusty 10 35.0

sid bid day22 101 10/10/9658 103 11/12/96

R1

S1 S2

Sailors(sid,sname,rating,age)Boats(bid,bname,color)Reserves(sid,bid,day)

LogicNotations

• $ Thereexists• " Forall• ∧ LogicalAND• ∨ LogicalOR• ¬NOT• ⇒ Implies

RelationalAlgebra(RA)

RelationalAlgebra

• Takesoneormorerelationsasinput,andproducesarelationasoutput– operator– operand– semantic– soanalgebra!

• Sinceeachoperationreturnsarelation,operationscanbecomposed– Algebrais“closed”


RelationalAlgebra• Basicoperations:

– Selection(σ)Selectsasubsetofrowsfromrelation– Projection(π)Deletesunwantedcolumnsfromrelation.– Cross-product(x)Allowsustocombinetworelations.– Set-difference(-)Tuplesinreln.1,butnotinreln.2.– Union(∪)Tuplesinreln.1orinreln.2.

• Additionaloperations:– Intersection(∩)– join⨝– division(/)– renaming(ρ)– Notessential,but(very)useful.


Projection

sname ratingyuppy 9lubber 8guppy 5rusty 10p sname rating S, ( )2

age35.055.5

page S( )2

• Deletesattributesthatarenotinprojectionlist.

• Schema ofresultcontainsexactlythefieldsintheprojectionlist,withthesamenamesthattheyhadinthe(only)inputrelation.

• Projectionoperatorhastoeliminateduplicates(Why)

– Note:realsystemstypicallydon’tdoduplicateeliminationunlesstheuserexplicitlyasksforit(performance)


S2

10DukeCS,Fall2016 CompSci516:DataIntensiveComputingSystems

Selection

s rating S>8 2( )

sid sname rating age28 yuppy 9 35.058 rusty 10 35.0

• Selectsrowsthatsatisfyselectioncondition

• Noduplicatesinresult.Why?

• Schemaofresultidenticaltoschemaof(only)inputrelation


S2

11DukeCS,Fall2016 CompSci516:DataIntensiveComputingSystems

CompositionofOperators

• Resultrelationcanbetheinputforanotherrelationalalgebraoperation– Operatorcomposition s rating S>8 2( )

sid sname rating age28 yuppy 9 35.058 rusty 10 35.0

sname ratingyuppy 9rusty 10

p ssname rating rating S, ( ( ))>8 2

Union,Intersection,Set-Difference

• Alloftheseoperationstaketwoinputrelations,whichmustbeunion-compatible:– Samenumberoffields.– `Corresponding’fieldshavethesametype

– sameschemaastheinputs

sid sname rating age22 dustin 7 45.031 lubber 8 55.558 rusty 10 35.044 guppy 5 35.028 yuppy 9 35.0

S S1 2È



S1 S2

Duke%CS,%Spring%2016 CompSci 516:%Data%Intensive%Computing%Systems 12


• Note:noduplicate– “Setsemantic”– SQL:UNION– SQLallows“bagsemantic”aswell:UNIONALL

sid sname rating age22 dustin 7 45.031 lubber 8 55.558 rusty 10 35.044 guppy 5 35.028 yuppy 9 35.0

S S1 2È



S1 S2



sid sname rating age31 lubber 8 55.558 rusty 10 35.0

S S1 2Ç

sid sname rating age22 dustin 7 45.0

S S1 2-



S1 S2


Cross-Product• EachrowofS1ispairedwitheachrowofR.• ResultschemahasonefieldperfieldofS1andR,withfield

names`inherited’ifpossible.– Conflict:BothS1andRhaveafieldcalledsid.

(sid) sname rating age (sid) bid day22 dustin 7 45.0 22 101 10/10/9622 dustin 7 45.0 58 103 11/12/9631 lubber 8 55.5 22 101 10/10/9631 lubber 8 55.5 58 103 11/12/9658 rusty 10 35.0 22 101 10/10/9658 rusty 10 35.0 58 103 11/12/96



sid bid day22 101 10/10/9658 103 11/12/96

RenamingOperator⍴

(sid) sname rating age (sid) bid day22 dustin 7 45.0 22 101 10/10/9622 dustin 7 45.0 58 103 11/12/9631 lubber 8 55.5 22 101 10/10/9631 lubber 8 55.5 58 103 11/12/9658 rusty 10 35.0 22 101 10/10/9658 rusty 10 35.0 58 103 11/12/96

§Ingeneral,canuse⍴(<Temp>,<RA-expression>)


(⍴sid →sid1S1)⨉ (⍴sid →sid1R1)or

⍴(C(1→sid1,5→sid2),S1⨉ R1)Cisthenewrelationname

Joins

• Resultschemasameasthatofcross-product.• Fewertuplesthancross-product,mightbeableto

computemoreefficiently

R c S c R S!" = ´s ( )

(sid) sname rating age (sid) bid day22 dustin 7 45.0 58 103 11/12/9631 lubber 8 55.5 58 103 11/12/96

S RS sid R sid1 11 1!" . .<


Findnamesofsailorswho’vereservedboat#103



Findnamesofsailorswho’vereservedboat#103

• Solution1:

• Solution2:

p ssname bid serves Sailors(( Re ) )=103 !"

p ssname bid serves Sailors( (Re ))=103 !"



ExpressinganRAexpressionasaTree



p ssname bid serves Sailors(( Re ) )=103 !"Sailors Reserves

σbid=103

⨝sid =sid

πsname

Alsocalledalogicalqueryplan

Findsailorswho’vereservedaredoragreenboat

• Canidentifyallredorgreenboats,thenfindsailorswho’vereservedoneoftheseboats:

r s( , ( ' ' ' ' ))Tempboats color red color green Boats= Ú =

p sname Tempboats serves Sailors( Re )!" !"

Can also define Tempboats using unionTry the “AND” version yourself



Useofrenameoperation

Whataboutaggregates?

• Extendedrelationalalgebra• 𝝲age,avg(rating)→ avgr Sailors• Alsoextendedto“bagsemantic”:allowduplicates

– Takeintoaccountcardinality– RandShavetupletresp.mandntimes– R∪ Shastm+n times– R∩Shastmin(m,n)times– R– Shastmax(0,m-n)times– sorting(τ),duplicateremoval(ẟ)operators



RelationalCalculus(RC)


RelationalCalculus• RAisprocedural

– πA(σA=a R)andσA=a (πAR)areequivalentbutdifferentexpressions

• RC– non-proceduralanddeclarative– describesasetofanswerswithoutbeingexplicitabouthowthey

shouldbecomputed

• TRC(tuplerelationalcalculus)– variablestaketuplesasvalues– wewillprimarilydoTRC

• DRC(domainrelationalcalculus)– variablesrangeoverfieldvalues


TRC:example

• Findthenameandageofallsailorswitharatingabove7

{P|∃ SϵSailors(S.rating >7⋀ P.sname =S.sname ⋀ P.age =S.age)}

• Pisatuplevariable– withexactlytwofieldssname andage(schemaoftheoutputrelation)– P.sname =S.sname ⋀ P.age =S.age givesvaluestothefieldsofananswer

tuple

• Useparentheses,∀ ∃ ⋁ ⋀ ><= ≠ ¬etc asnecessary• A⇒ Bisveryusefultoo

– nextslideDukeCS,Fall2018 CompSci516:DatabaseSystems 30


$ Thereexists

A⇒ B

• A“implies”B• Equivalently,ifAistrue,Bmustbetrue• Equivalently,¬A⋁ B,i.e.

– eitherAisfalse(thenBcanbeanything)– otherwise(i.e.Aistrue)Bmustbetrue


UsefulLogicalEquivalences

• "xP(x) =¬$x [¬P(x)]

• ¬(P∨Q) =¬P∧ ¬Q• ¬(P∧ Q) =¬P∨ ¬Q

– Similarly,¬(¬P∨Q)=P∧ ¬Qetc.

• AÞ B=¬A∨ B


$ Thereexists" Forall∧ LogicalAND∨ LogicalOR¬ NOT

deMorgan’slaws

TRC:example

• Findthenamesofsailorswhohavereservedatleasttwoboats



TRC:example

• Findthenamesofsailorswhohavereservedatleasttwoboats

{P|∃ SϵSailors(∃ R1ϵ Reserves∃ R2ϵReserves(S.sid =R1.sid⋀ S.sid =R2.sid⋀ R1.bid≠R2.bid)⋀ P.sname =S.sname)}



TRC:example

• Findthenamesofsailorswhohavereservedallboats• Calledthe“Division”operation



TRC:example

• Findthenamesofsailorswhohavereservedallboats• DivisionoperationinRA!

{P|∃ SϵSailors[∀BϵBoats(∃ Rϵ Reserves(S.sid =R.sid ⋀R.bid =B.bid))]⋀ (P.sname =S.sname)}



TRC:example

• Findthenamesofsailorswhohavereservedallred boats



HowwillyouchangethepreviousTRCexpression?

TRC:example

• Findthenamesofsailorswhohavereservedallred boats{P|∃ SϵSailors(∀BϵBoats(B.color =‘red’⇒ (∃ RϵReserves(S.sid =R.sid ⋀ R.bid =B.bid)))⋀ P.sname =S.sname)}

RecallthatA⇒ Bislogicallyequivalentto¬A⋁ Bso⇒ canbeavoided,butitiscleanerandmoreintuitive



DRC:example

• Findthenameandageofallsailorswitharatingabove7

TRC:{P|∃ SϵSailors(S.rating >7⋀ P.name =S.name ⋀ P.age =S.age)}

DRC:{<N,A>|∃ <I,N,T,A>ϵSailors⋀ T>7}

• Variablesarenowdomainvariables• WewilluseuseTRC

– bothareequivalent• Anotheroptiontowritecomingsoon!



MoreExamples:RC

• Thefamous“Drinker-Beer-Bar”example!


Acknowledgement:examplesandslidesbyProfs.BalazinskaandSuciu,andthe[GUW]book

UNDERSTANDTHEDIFFERENCEINANSWERSFORALLFOURDRINKERS

DrinkerCategory1

Find drinkers that frequent some bar that serves some beer they like.

Likes(drinker, beer)Frequents(drinker, bar)Serves(bar, beer)

41CompSci516:DatabaseSystemsDukeCS,Fall2018

DrinkerCategory1


Q(x) = $y. $z. Frequents(x, y)∧Serves(y,z)∧Likes(x,z)

a shortcut for{x | $Y ϵ Frequents $ Z ϵ Serves $ W ϵ Likes ((T.drinker = x.drinker) ∧(T.bar = Z.bar) ∧ (W.beer =Z.beer) ∧ (Y.drinker =W.drinker) }



Thedifferenceisthatinthefirstone,onevariable=oneattributeinthesecondone,onevariable=onetuple(TupleRC)Bothareequivalentandfeelfreetousetheonethatisconvenienttoyou

DrinkerCategory2


Find drinkers that frequent only bars that serves some beer they like.



43CompSci516:DatabaseSystems

Q(x) = …

DukeCS,Fall2018

DrinkerCategory2




Q(x) = "y. Frequents(x, y)Þ ($z. Serves(y,z)∧Likes(x,z))



DrinkerCategory3



Find drinkers that frequent some bar that serves only beers they like.





Q(x) = …

DukeCS,Fall2018

DrinkerCategory3






Q(x) = $y. Frequents(x, y)∧"z.(Serves(y,z) Þ Likes(x,z))



DrinkerCategory4



Find drinkers that frequent only bars that serves only beer they like.







Q(x) = …

DukeCS,Fall2018

DrinkerCategory4



Find drinkers that frequent only bars that serves only beer they like.




Q(x) = "y. Frequents(x, y)Þ "z.(Serves(y,z) Þ Likes(x,z))



48DukeCS,Fall2018 CompSci516:DatabaseSystems

WhyshouldwecareaboutRC• RCisdeclarative,likeSQL,andunlikeRA(whichisoperational)

• Givesfoundationofdatabasequeriesinfirst-orderlogic– youcannotexpressallaggregatesinRC,e.g.cardinalityofarelationorsum(possibleinextendedRAandSQL)

– stillcanexpressconditionslike“atleasttwotuples”(oranyconstant)

• RCexpressionmaybemuchsimplerthanSQLqueries– andeasiertocheckforcorrectnessthanSQL– powertouse" and Þ– then you can systematically go to a “correct” SQL

query


FromRCtoSQL

Q(x) = $y. Likes(x, y)∧"z.(Serves(z,y) Þ Frequents(x,z))

Query: Find drinkers that like some beer (so much) that they frequent all bars that serve it

CompSci516:DatabaseSystems 50


DukeCS,Fall2018

FromRCtoSQL

Q(x) = $y. Likes(x, y)∧"z.(Serves(z,y) Þ Frequents(x,z))

Query: Find drinkers that like some beer so much that they frequent all bars that serve it

Step 1: Replace " with $ using de Morgan’s Laws

Q(x) = $y. Likes(x, y)∧ ¬$z.(Serves(z,y) ∧ ¬Frequents(x,z))



"x P(x) same as¬$x ¬P(x)

¬(¬P∨Q) same asP∧ ¬ Q

º Q(x) = $y. Likes(x, y)∧"z.(¬ Serves(z,y) ∨ Frequents(x,z))

DukeCS,Fall2018

FromRCtoSQL

SELECT DISTINCT L.drinkerFROM Likes LWHERE not exists

(SELECT S.barFROM Serves SWHERE L.beer=S.beer

AND not exists (SELECT * FROM Frequents FWHERE F.drinker=L.drinker

AND F.bar=S.bar))



DukeCS,Fall2018

Q(x) = $y. Likes(x, y) ∧¬ $z.(Serves(z,y)∧¬Frequents(x,z))

Step 2: Translate into SQL

Query: Find drinkers that like some beer so much that they frequent all bars that serve it

Wewillseea“methodicalandcorrect”translationtrough“safequeries”inDatalog

Summary

• YoulearntthreequerylanguagesfortheRelationalDBmodel– SQL– RA– RC

• Allhavetheirownpurposes

• Youshouldbeabletowriteaqueryinallthreelanguagesandconvertfromonetoanother– However,youhavetobecareful,notall“valid”expressionsinonemay

beexpressedinanother– {S|¬(SϵSailors)}– infinitelymanytuples– an“unsafe”query– Morewhenwedo“Datalog”,alsoseeCh.4.4in[RG]


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Relational Algebra and Relational Calculus - cs.duke.edu · Recap: SQL --Lecture 2/3 •Creating/modifying relations •Specifying integrity constraints •Key/candidate key, superkey,