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A.1 Relations, Orders, and Lattices

A.1 Relations, Orders, and Lattices

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Transitive.IfaRbandbRc,thenaRc.

Reflexive.Foranya,wehaveaRa.

Antisymmetric.IfaRbandbRa,thena==b.

Aniceintuitiveexampleofanorderis ontheintegers.

Wesaythatanorderistotalwhen,foranydistinctx,y,we

haveeitherxRyoryRx.(Again,thinkof ontheintegers.)In

contrast,apartialorderdoesnotrequirethatalldistinctpairs

beorderable.Somepairscanbeincomparable.

"IloveeachofmydaughtersmorethanIlovetheraccoon

thatlivesbehindthegarage."

"ButIcan'ttellyouwhetherIlovemyolderdaughtermore

thanIlovemyyoungerdaughter."

(Ifonewantstobepedanticaboutit,atotalorderismerelya

specialcaseofapartialorder.)

Inapartialorder,anelementcistheleastupperboundof

elementsaandbifcsatisfiestheseproperties:

aRc.

bRc.

IfaRdandbRd,thencRd.

Intuitively,thefirsttwoconditionsmeanthatcisindeedan

upperboundofaandb;thethirdconditionmeansthatitis

"smaller"thananyotherupperboundofaandb.Wecan

similarlydefineagreatestlowerbound.

Alatticeisapartiallyorderedsetsuchthatanytwoelements

havealeastupperboundandagreatestlowerbound.Infinite

latticescanbeabitstrangetothinkabout.However,suppose

insteadthatwehaveafiniteone,builtfromtakingthe

transitiveclosureofasimplerelation,andwedrawitasa



directedgraph:

Anodeforeachelementoftheset

Anedgeforeachpairunderthesimplerelation

Pathsforthetransitiverelation

Theresultingstructurewillactuallylookabitlikewhatwecalla

latticeinordinaryEnglish.Eachpairofnodesneedstohavea

uniqueleastupperbound,whichforcesustohavearegular

lattice-workstructure.Thegraphitselfwillhavedistinctivetop

andbottomelements.FigureA.1sketchestheseconcepts.

FigureA.1.Thisdirectedgraphisanexampleofasimple

lattice,wheren1 n2ifandonlyifadirectedpathexists

fromn1ton2.Weseethatnodeeisthegreatestlower

boundofnodesbandc.Nodese,h,i,andjarealllower

boundsofthesetwonodes,buth e,i e,andj e.



A.2Functions

AmathematicalfunctionffromasetDtoasetRisamapthat

takeseachelementxϵDtosomeelementf(x)ϵR.Apartial

functioncanleavesomef(x)undefined.ThesetDhereis

calledthedomainoff;thesetRisitsrange.Sometimes,this

isallexpressedmathematicallyas

f:D



R.



Sometimes,thevaluef(x)ϵRissaidtobetheimageofx

underf;thevaluexissaidtobethepreimageofthevaluef

(x).

TherearenorequirementsthatDandRbedistinct;folksoften

talkaboutjustoneset(e.g.,"afunctionontheintegers")when

D=R.

Wesometimesdistinguishsomespecialpropertiesoffunctions.

Afunctionfisonetoone(orinjective)ifforanyx yϵD,

wehavef(x) f(y).Eachxgoestoitsveryownimage.

Afunctionfisonto(orsurjective)whenfhitsevery

elementinitsrangeR.

Afunctionfthatisbothinjectiveandsurjectiveisa

bijection.

Thewordhomomorphicderivesfromthephrasesameform.

Afunctionishomomorphicwhenitpreservessome

interestingformofitsdomainset,suchassomespecial

relations.

Wemightblunderandsomehowmanagetointroduceanfsuch

thatit'snotclearwhichelementofRsomexmapsto.Inthis

case,ourfunctionfisnotwelldefined.



A.3ComputabilityTheory

Givenafunctionf:D R,ourfirstreaction,ascomputer

people,mightbetothinkabouthowwewouldcomputeit:how

wewouldwriteaprogramthattakesxϵDasaninputandspits

outf(x).

Thefieldofthetheoryofcomputationdevelopedprecise

mathematicalcharacterizationsofwhatthings"programs"can

do.Thischaracterizationleadstoasurprising,counterintuitive

result:Thereexistwell-definedfunctionsthatcannotbe

computed.

Wequicklypresentthisresult.



A.3.1ThingsCanBeUncountable

Westartwithcountability.

Recallthesetofnaturalnumbers:

={1,2,...}.

Theset isclearlyinfinite;othersetsareinfiniteaswell.

Mathematically,wesaythatasetSiscountablewhenwecan

putSinone-to-onecorrespondencewith .Onemightask:

Arethereanyinfinitesetsthatarenotcountable?Theansweris

yes.Forexample,considertherealnumbersbetween0and1:

R={rϵ :0


SinceeachelementofRisbetween0and1,wecanwriteeach

indecimalas0,followedbyadecimalpoint,followedbysome

numberofdigits,perhapsanonterminatingsequenceofthem.

We'llalsodefineaTwist()functionthattwistsadigittobe

somethingverydifferent:



WecanshowthatRisnotcountable,usingsomethingcalled

Cantor'sdiagonalizationtechnique.IfRwerecountable,we

couldenumerateitasr1,r2,...andhittheentireset.Wecould

usethisenumerationandthedecimalrepresentationtobuilda

table:Rownwouldcontainrn,andcolumnkwouldcontainthe

kthdecimaldigit.(SeeFigureA.2.)

FigureA.2.Iftherealnumbersbetween0and1were

countable.wecouldenumeratethemasr1,r2....



Fromthistable,wecanconstructanotherelementofR(callit

s)bygoingalongthediagonalbuttwistingthingsateachstep

—weletthenthdigitofsbeTwist(d),wheredisthenthdigit

ofrn.(SeeFigureA.3.)ThisnewnumbersisclearlyinR.Butit

alsocannotbeinthistable—forifs=rmforsomem,what

wouldthemthdigitofsbe?Cool,eh?

FigureA.3.Bymovingalongthediagonalandtwisting



thedigittheresoit'sdifferent,weproducearealnumber

sthatcannotbeinthisenumeration.



A.3.2ThingsCanBeUncomputable

Wecanusethesameprincipletoshowthatthereare

uncomputablefunctions.LetPbethesetofpossibleprograms

andIbethesetofpossibleinputstotheseprograms.Whenwe

getseriousaboutformalizingPandI,weendupwithsetsthat

arecountable.So,wecanusethesefactstobuildatable:Row

ntalksaboutprogrampn;columnktalksaboutwhathappens

whenwerunprogramsoninputik.Inentryn,k,wewriteHalt

orNotHalt,dependingonwhetherPn(ik)haltsorrunsforever.

(SeeFigureA.4.)

FigureA.4.BecausetheTuringmachinesarecountable,

wecanenumeratethem(downtherowshere).Because

theirpossibleinputsarecountable,wecanenumerate

themaswell(acrossthecolumnshere).Eachmachineon

eachinputeitherhaltsordoesn't;inthistable,weshow

whathappens.



WecandefineanewTwist()function:



Fromthistable,wecanconstructafunctionfbygoingalong

thediagonalbuttwistingthingsateachstep:

f(in)=Twist(pn(in)).



(SeeFigureA.5.)

FigureA.5.Bymovingalongthediagonalandtwisting

theentrytheresoit'sdifferent,weproduceafunctionf

thatcannotbecomputedbyaTuringmachineinthis

enumeration.



Semantically,thisfunctionfreturnsHaltoninexactlywhenPn

doesnothaltoninputin.Consequently,noprogramcan

computef!Allpossibleprogramsappearinthistable.So,iff

werecomputable,itmustbecomputedbypmforsomem.

However,howcanpm(im)behavethesameasf(im)?Ifpm(im)

halts,thenf(im)mustsayNotHalt,sopm f;ifpm(im)does

nothalt,thenf(im)musthalt,sopm f.Cool,eh?

Theuncomputableproblemofdeciding,yesorno,whethera

givenprogramhaltsonagiveninputiscalledtheHalting

Problem.Onecanusethisideaasaspringboardintoaworldof

fascinatingtheoreticalandpracticalworkanduncomputable

functions.(Sipser'sIntroductiontotheTheoryofComputation

coverstheseproblemsindetail[Sip97].)Asayoungpostdoc,

oneofuswasonceaskedtowriteaprogram(fortheU.S.

PostalService)thatessentiallysolvedtheHaltingProblem.

CitingAlanTuring'sclassic1930spaper[Tur37]inourfinal

report[Smi05]wascauseformuchmirth(atleastamongthe



authorsofthereport).



A.4Frameworks

SectionA.3showedhowfunctionsexistthatcannotbe

computed.Thisexistencecanbeusefulwhenwewanttotalk

rigorouslyabouthardtasks:Forexample,wecanshowthatin

ordertodothistask,wehavetosolvetheHaltingProblem.This

isanexampleofacomputation-theoreticargumentframework.

Theoryalsogivesusotherframeworkstoreasonaboutthings.

Amongcomputablefunctions,wecanstartrigorouslyreasoning

aboutthetimeandspaceresources(andtypesof

computationaldevices)ittakestocomputethem.Thistheory

givesusawaytotalkaboutthecomplexityofcomputable

problems.Informally,wesayaproblemhascomplexityO(f

(n))iftheresourcesitrequirestosolveaninstanceofsizen

growsbynotmorethanf(n),asymptotically.Aproblemhas

complexity (f(n))iftherequiredresourcegrowsbyatleastf

(n)asymptomatically.(Notehowever,thatcomplexitytheory

usuallyassumesthattheproblem'sparametersarewritten

downinbinary.)

Theresultingframeworkgivesanotherwaytotalkabouthard

tasks.Forexample,wecanshowthatdoingthistaskrequires

doingafunctionthat,althoughcomputable,takesmore

resourcesthanisfeasible.Thisisanexampleofacomplexitytheoreticargumentframework.

Manyaspectsofcryptographyareevenweakerthan

complexity-theoretic.Foroneexample,noclearcomplexitytheoreticboundunderliesthesecurityofSHA-1.Foranother

example,everyonesuspectsthatfactoringishard,butnoone

reallyknows;furthermore,breakingRSAmightbeeveneasier.

Wealsothussometimesseeatacit"cryptographic"framework

—whereonesimplyhopesthatcertainkeyfunctionsare

intractable.(Wethenarguethatbreakingourschemerequires

breakingoneofthesehopefullyintractablefunctions.)

Chapter7discussedyetanotherframework:theuseof



informationtheorytocharacterizetherawinformationpresent

inmessage.Iftheinformationisn'tthere,thentheadversary

can'tgetit.However,iftheinformationisthere,thatdoesn't

meanthattheadversarycangetit.Whenitcomesto

consideringcomputationintherealworld,informationtheory

hasaglaringdeficiency—itneglectstotakeintoaccountthe

feasibility(oreventhepossibility)ofactuallyextractingthe

informationfromthemessage.



Conundrum

Asathoughtexerciseabouttheseframeworks,supposethat

Alicewantstosenda20-characterASCIImessageMtoBob.

Shecanchoosemanyencodings:

M1.SheencodeseachbitofMasaninstanceofthe

satisfiabilityproblem,overanumberofvariablespolynomial

in|M|.(Satisfiablemeansthatthebitis1.)

M2.SheencodeseachbitofMasaninstanceoftheHalting

Problem.(Haltsmeansthatthebitis1.)

M3.SheencryptsthemessagewithTDES.

M4.Shesendsthemessageinplaintextbutdeletesthe

least-significantbitofeachbyte.

(Thesatisfiabilityproblemcomesfromcomplexitytheory:One

isgivenaBooleanformulaconsistingofANDsandORsof

variablesandtheirnegationsandneedstoanswer:Istherea

truthassignmenttothevariablesthatsatisfiesthisformula?)

HowmuchinformationaboutMispresentineachofAlice's

formats?InformationtheorytellsusthatM1,M2,andM3

containalltheinformation.However,decodingM1isintractable,

anddecodingM2is,ingeneral,notevencomputable.IfAlice

wantstominimizeinformationexposedtoaneavesdropper,she

shouldchooseM4—eventhoughthatgivesawayhalfthe



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