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Forbidding and enforcing of formal languages, graphs, and partially ordered sets

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Forbidding and enforcing of formal languages, graphs, and partially ordered sets
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Genova, Daniela
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Classes of formal languages
Language families
Subwords
Subgraphs
Posets
DNA computing
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ABSTRACT: Forbidding and enforcing systems (fe-systems) provide a new way of defining classes of structures based on boundary conditions. Forbidding and enforcing systems on formal languages were inspired by molecular reactions and DNA computing. Initially, they were used to define new classes of languages (fe-families) based on forbidden subwords and enforced words. This paper considers a metric on languages and proves that the metric space obtained is homeomorphic to the Cantor space. This work studies Chomsky classes of families as subspaces and shows they are neither closed nor open. The paper investigates the fe-families as subspaces and proves the necessary and sufficient conditions for the fe-families to be open. Consequently, this proves that fe-systems define classes of languages different than Chomsky hierarchy. This work shows a characterization of continuous functions through fe-systems and includes results about homomorphic images of fe-families.This paper introduces a new notion of connecting graphs and a new way to study classes of graphs. Forbidding-enforcing systems on graphs define classes of graphs based on forbidden subgraphs and enforced subgraphs. Using fe-systems, the paper characterizes known classes of graphs, such as paths, cycles, trees, complete graphs and k-regular graphs. Several normal forms for forbidding and enforced sets are stated and proved. This work introduces the notion of forbidding and enforcing to posets where fe-systems are used to define families of subsets of a given poset, which in some sense generalizes language fe-systems. Poset fe-systems are, also, used to define a single subset of elements satisfying the forbidding and enforcing constraints. The latter generalizes graph fe-systems to an extent, but defines new classes of structures based on weak enforcing. Some properties of poset fe-systems are investigated. A series of normal forms for forbidding and enforcing sets is presented.This work ends with examples illustrating the computational potential of fe-systems. The process of cutting DNA by an enzyme and ligating is modeled in the setting of language fe-systems. The potential for use of fe-systems in information processing is illustrated by defining the solutions to the k-colorability problem.
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Dissertation (Ph.D.)--University of South Florida, 2007.
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by Daniela Genova.
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by DanielaGenova Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:NatasaJonoska,Ph.D. GregoryMcColm,Ph.D. MasahicoSaito,Ph.D. StephenSuen,Ph.D. DateofApproval: June14,2007 Keywords:Classesofformallanguages,Languagefamilies,Subwords,Subgraphs,Posets,DNAcomputing cCopyright2007,DanielaGenova

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Iwouldliketothankallmembersofmydissertationcommittee:G.L.Mc-Colm,M.Saito,andS.Suenfortheirtimeandeortinreviewingpreviousdraftsofthisdissertationandprovidingvaluablesuggestions,aswellas,greatideasforfuturework.Iamveryproudtohavesuchoutstandingscholarsonmydissertationcommittee! Myspecialthanksgotoallmymathematicsprofessorswhonourishedmyloveformathematicsandencouragedmetopursuechallengingideas.IthankN.Jonoska,G.McColm,S.Suen,M.Saito,M.Ismail,J.Liang,E.Clark,B.Curtin,andB.Shektmanforthat.Inaddition,thisjourneywouldnothavebeenpossiblewithoutthehelpfromtheentireMathematicsDepartmentatUSFledbyM.McWatersandS.RimbeyandthesupportstaJim,Aya,Sarina,Evelyn,Beverly,Denise,MaryAnn,Nancy,Frances,Barbara.Ithankthemall! Twopeoplewereveryinstrumentaltoenhancingmyteachingexperience:F.

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Theconstantsupport,encouragement,andlovefromfamilyandfriendswasinvaluable.IthankPhilip,Stella,Laura,myparents,Zornitza,LauraK.,John,Karen,Elena,andPeterforalwaysbelievinginmeandprovidingpurpose. Lastbutnotleast,Iwouldliketorecognizethefollowinginstitutionsfortheirnancialsupport.IthanktheDapartmentofMathematicsattheUniversityofSouthFloridaforawardingmeaGraduateTeachingAssistantship;EckerdCollegeforprovidingmewithaVisitingAssistantProfessorshipfortwoyears,whilebeingagraduatestudent;theGSBOforawardingmetravelgrantstopresentmyresearchatlocal,state,andinternationalconferences;UniversityofUtahandtheNationalScienceFoundationforthetravelgranttopresentmyresearchinSydney,Australia;theAssociationofWomeninMathematicsforsponsoringmytriptotheJMMinNewOrleansandmakingitpossibleformetoparticipateintheAWMworkshopandpresentmyresearchthere;theDepartmentsofMathematicsatIndiana-PurdueUniversityatFortWayneandUniversityofNorthFloridaforinvitingmetopresentmyworkthere.Ican'thelpbutrecognizehowfortunateIamtohavehadtheirsupport!4

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Abstractiv 1Introduction1 2ForbiddingandEnforcingofFormalLanguages6 2.1Denitions................................6 2.2MinimalNormalForms.........................9 2.3MaximalLanguages...........................12 2.4ExtendedForbiddingSets.......................20 2.5GeneratedLanguages..........................23 3TopologicalPropertiesoffe-FamiliesofLanguages28 3.1TheCantorSpaceP(A)........................28 3.2ContinuousFunctions..........................32 3.3ChomskyFamiliesasSubspacesofP(A)...............34 3.4TopologicalPropertiesoffe-Families.................35 4Morphismsandfe-FamiliesofLanguages41 4.1MorphicMapsandfe-Families....................41 4.2CharacterizingMorphicImagesasfe-Families............44 5ForbiddingandEnforcingofGraphs49 5.1Denitions................................49i

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5.3Graphfe-SystemsandTheirProperties...............64 5.4ForbiddingthroughEnforcing.....................68 5.5CharacterizationsofSomeClassesofGraphsbyfe-Systems....69 6NormalFormsforGraphfe-Systems75 6.1NormalFormsforForbiddingSets...................75 6.2NormalFormsforEnforcingSets...................88 7ForbiddingandEnforcingonPartiallyOrderedSets94 7.1fe-FamiliesasSetsofSubposets....................95 7.2fe-SystemsDeningaSingleSubposet................103 7.3UpperBounds..............................108 7.4NormalFormsforForbiddingSets...................111 7.5NormalFormsforEnforcingSets...................119 8Computingwithfe-Systems123 8.1ModelingMolecularBondingandSplicingSystems.........124 8.2InformationProcessingbyfe-Systems................127 References132 AbouttheAuthorEndPageii

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2.2TreesassociatedwiththemaximallanguagesfromExample2.3.2.19 3.1Continuousfunctions..........................33 5.1ThegraphSC3P4C4fromExample5.2.3................52 5.2MinimalconnectinggraphsofFfromExample5.2.5........53 5.3GraphsrelatedtoExample5.2.20...................56 5.4GraphsrelatedtoExample5.2.29...................60 5.5ThegraphGUfromDenition5.2.30.................61 5.6Aminimalconnectinggraphof~XinGfromExample5.2.34....63 5.7GraphrelatedtoExample5.3.6....................65 5.8TreerelatedtoProposition5.5.1....................70 5.9GraphsrelatedtoEinCorollary5.5.8.................74 6.1GraphsrelatedtoEinExample6.2.8.................90 8.1(a)Cuttingwithrestrictionenzymes(b)DNArecombination....125 8.2Missingnucleotides:(a)diagonally(b)inonestrand.........126iii

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ChemicalpropertiesofDNAandactionsofrestrictionenzymes(see[37,42])haveinspiredmanyDNAcomputingmodelslike[1,22,24,41].EncodingtheproblemusingDNAmoleculesinvolvesavoidingundesirablehybridizationofDNAstrands.DNAcodingpropertiesneededtoproperlyencodeaproblemhavebeenintroducedin[21]andwidelystudiedinrecentyears(seeforex.[23,28]). Allofthesemodelsarebasedonclassicalformallanguagetheory(see[20,36])1

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Chapters2,3,and4areanextendedversionof[16].Thebasictopologicalnotionsareassumedandcanbefoundin[29].Thedenitionsoffe-systemsarestatedalongwithsomeoftheirproperties,includingnewnormalforms.Theorem2.4.3providesacharacterizationofextendedf-families.TheminimalgeneratedlanguagesextendthenotionofE-extensionsintroducedin[8].Thewordmetric3.1.1onthespaceofformallanguagesisthesameastheoneimplicitlyusedin[10,40]andfollowsasimilarapproachasin[26,27].ThispapershowsthatthelanguagespaceequippedwiththewordmetricishomeomorphictotheCantorspace.Acharacterizationofthecontinuousmorphismsonthatspaceisprovided.Theorem3.2.1statesthatamorphismiscontinuousifandonlyifitis-free.Thiscorrespondstothecharacterizationofcontinuousmapsoninnitesequencesin[39].Inaddition,examplesofothercontinuousfunctionsarepresentedthatcomefromwellknownoperationsonlanguagessuchastakingproductsoflanguageswithaxedlanguage,2

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InChapter5anentirelynewwayofdeningclassesofgraphsispresentedbasedonboundaryconditions.Thechapterintroducesanewnotioncalledconnectinggraphs.Givenasetofconnectedgraphs,aconnectinggraphisagraphwhichcon-tainseachgraphfromthesetasasubgraph.Minimalconnectinggraphsaredenedwithrespecttosubgraphs.Thisworkshowsthatevensmallsetsofgraphshaveaninnitenumberofminimalconnectinggraphs.Thischapterdenesforbiddingsetsofgraphsasacollectionofnitesetsofconnectedgraphs.Thef-familiesaredenedasallgraphsthatdonothaveforbiddencombinationsofsubgraphs.His-torically,forbiddengraphshaveonlybeendenedas\strict"forbiddingsets,whereeachforbidderisasingleton.Suchdenitionhasonlybeenusedinthecaseofin-ducedsubgraphsratherthansubgraphs.In[7,12,17])themainobjectiveoftheseforbiddengraphsistoprovehamiltonicity.Forbiddengraphswere,also,usedinextremalgraphtheoryinTurantypeproblems.Foracomprehensivelistofpaperssee[2,18].Inthiswork,e-familiesaredenedasgraphsinwhichcertainsubgraphsarerequiredtobe\enclosed"inlargersubgraphs.AnewtoolforrepresentingthesetofconnectedgraphscalledtheGUgraphisintroduced.Itisusedtorepresent3

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Chapter7extendstheforbiddingandenforcingparadigmtopartiallyorderedsets.Intherstsectionthefe-familyisdenedasasetofsubposetsandinthisrespect,itgeneralizesthelanguagefe-systems.Severalnormalformsforforbiddingsetsandenforcingsetsfrom[8,9,40]aregeneralizedtoposets,includingtheminimalnormalformandnitarynormalform.TheremainingsectionsofChapter7deneandstudythefe-familyasasingleposetandinthisrespectgeneralizesthegraphmodelfromChapter5toanextent.Forbiddingsetsareageneralizationofthegraphforbiddingsystems,buttheenforcingsetspresentanewwaytodeneclassesofstructuresusingtheconceptof\weak"enforcing.Examplesofdierenttypesofposets,suchasthenaturalnumberswithdivisibilityandwordswithsubwordorderarepresented.Suchfe-systemsareusedtocharacterizesomefamiliarclassesofstructures.Inthecaseofwords,thefe-systemsdeneasinglelanguage,asopposedtothelanguagefe-systemsdiscussedinChapter2,whereafamilyoflanguagesisobtained.Suchapproachofdeninglanguagesisentirelynewcomparedtothetraditionalwaystodenealanguageusingagrammaroranautomatonsee[36].Again,somenormalformsforforbiddingsetsandenforcingsetsarepresented. Chapter8isdevotedtothemotivationforandexamplesofapplicationsoftheforbiddingandenforcingtheory.TherstsectiondiscussesanexampleofsplicingwithanenzymeandligatingDNAstrands,whichprovesthatfe-systemscanprovideanequivalentdenitionofsplicing,originallydenedin[19].Thesecondexampleshowshowfe-systemscanbeusedforinformationprocessing.Here,deningthesolutionstothek-colorabilityproblemisusedasanexample. Thispaperisorganizedasfollows.Chapter2deneslanguagefe-systemsandpresentssomeoftheirproperties.Itinvestigatesnormalformsforfe-systems,max-imallanguagesinforbiddingfamilies,extendedforbiddingsets,andgeneratedsets4

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x,ifthereexists;t2Asuchthatx=syt.Thesetofallsubwordsofawordxisdenotedbysub xgandthesetofallsubwordsofwordsinalanguageLbysub Thedenitionsforforbiddingandenforcingsystemsthatfollowarefrom[8,40]withtheexceptionofextendedforbiddingsets.Denition2.1.1 Aforbiddingset^Fiscalledextendedifitsforbiddersarenotnecessarilynite. AlanguageLissaidtobeconsistentwithaforbidderF,denotedbyLcon F,ifF6sub FforallF2F.IfLisnotconsistentwithF,thenotationisLncon ForaforbiddingsetF,thefamilyofF-consistentlanguages(theF-family)isL(F)=fLjLcon ThefamilyL(F)issaidtobedenedbytheforbiddingsetF.AfamilyoflanguagesLisaforbiddingfamily(f-family),ifthereisaforbiddingsetFsuchthatL=L(F).Twoforbiddingsetsareequivalentiftheydenethesamefamilyoflanguages.Theequivalencerelationisdenotedby.Inotherwords,FF0ifandonlyifL(F)=L(F0).Example2.1.2 Notethattheemptylanguage;andfgareinL(F)foreveryF(see[40]).Thenextremarksimplysaysthatifnothingisforbidden,theneverythingisallowedandviceversa.Remark2.1.3

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AlanguageLsatisesanenforcer(X;Y),denotedLsat ForanenforcingsetE,thefamilyofE-satisfyinglanguagesisL(E).ThefamilyoflanguagesdenedbytheenforcingsetEisL(E)=fLjLsat ObservethateveryL(E)containsthelanguageA. Inbothforbiddingandenforcing,itisassumedthatthelanguagesunderconsid-erationcontainwordsoveraxednitealphabetA.Ifthealphabetisnotspecied,thenitisassumedthatAisthesetofallsymbolsthatappearinthewordsofthesetofforbiddersand/orintheenforcingset. ThedenitionforenforcersallowsthesetXinanenforcer(X;Y)tobeempty.Inthiscase,foreverylanguageL,Lsat

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TheenforcingsetEistrivialifEisemptyorEcontainstrivialenforcersonlyandnon-trivialotherwise.Thus,Eisnon-trivialifandonlyifL(E)6=P(A).Inwhatfollows,unlessotherwisestated,allenforcersarenon-trivial.Denition2.1.7 FromthedenitionsandobservationsintheprevioussectionitfollowsthatthereisnoforbiddingsetF,forwhichL(F)isempty.Also,thereisnoenforcingsetE,suchthatL(E)=;.ThenextremarkisusedintheproofofTheorem3.4.9.Remark2.2.1

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ThefollowingpropositioncharacterizesthelanguagesL,forwhichthereexistsanontrivialforbiddingsetFsuchthatL2L(F).Proposition2.2.4 Proof.Ifsub NotethatforeverylanguageL6=;thereisanontrivialEsuchthatL2L(E).Forexample,letE=f(;;fwg)jw2Lg. AforbiddingsetFissaidtobeinminimalnormalformifFissubwordfree,i.e.,awordinaforbiddercannotbeasubwordofanotherwordinthesameforbidder,andsubwordincomparable,i.e.,foranytwoforbiddersF1andF2,sub

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AnenforcingsetEissaidtobenitary,ifforeachX2E(1),thereisanitenumberofenforcers(X;Yi)inE.Thefollowingtheoremisfrom[8].Theorem2.2.6 Papers[8,9,40]discussnormalformsofforbiddingsetsandnormalformsofenforcingsets.Itturnsoutthateventhoughaforbiddingsetmaybegiveninminimalnormalformandanenforcingsetmaybegiveninanitarynormalform,thefe-systemasawholemaystillberedundant.Theremainderofthissectionpresentssuchobservations.Proposition2.2.8 Proof.Let(F;E)begiven.LetF2Fand(X;Y)2EbesuchthatFX.SinceE0E,fromRemark2.2.7itfollowsthatL(F;E)L(F;E0).LetL2L(F;E0).SinceLcon Thecorollarybelowpointsoutthatcertainenforcingsetsmayberemovedfromthefe-systemwithoutchangingthefe-family.Corollary2.2.9

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F.Therefore,X6L,i.e.,Lsat Proof.Let(F;E)begivenandletE0bedenedasintheconditionsoftheproposi-tion.ItisclearthatL(F;E0)L(F;E).AssumeL2L(F;E)andlet(X;Y0)2E0.Then,thereisan(X;Y)2EsuchthatY0=Yn([F2FYF)andLsat Proof.Obviously,L(F;E)L(F;E0).LetL2L(F;E0)andlet(X;Y)2EnE0.Then,thereisanF2FsuchthatXF6=;.SinceF6sub

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ThesetP(A)withinclusionoflanguagesformsapartiallyorderedsetdenotedby(P(A);).EverychainCinP(A)containsanupperbound,namelyA,andbyZorn'slemmaP(A)hasamaximalelement.ForafamilyoflanguagesL,thesetofitsmaximallanguagesisdenotedbyM(L).Thesetofmaximallanguagesofanf-familyL(F)isdenotedbyM(F).NotethateverychaininL(F),e.g.,L1L2:::isboundedby[i1Li(see[8]),whichimpliesthatforeveryL2FthereisaLmax2L(F)suchthatLLmax. Theexamplebelowhasbeenconsideredin[8,9,40].Example2.3.2 Alanguageiscalledfactorial(closedbyitsfactors)ifitcontainsallofitsfactors(subwords)(see[36]).Ingeneral,Lsub Proof.LetFbeaforbiddingsetandletLbeamaximallanguageinL(F).(i)ItisclearthatLsub

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Proof.(i)AssumethatM(F)L(F0).LetL2L(F).Then,thereisamaximallanguageLmax2L(F),i.e.,Lmax2M(F),suchthatLLmax.SinceLmax2L(F0),fromLemma2.3.4itfollowsthatL2L(F0).Hence,L(F)L(F0).(ii)AssumeL(F)=L(F0).Then,ifL2M(F)itfollowsthatL2L(F),whichimpliesthatL2L(F0).Hence,thereisaLmax2M(F0)suchthatLLmax.SinceLmaxisalsoinL(F)andL2M(F),itfollowsthatL=Lmax.Hence,M(F)M(F0).Similarly,M(F0)M(F);therefore,M(F)=M(F0).Conversely,assumethatM(F)=M(F0)andletL2L(F).Then,thereisaLmax2M(F)withLLmax.SinceLmax2M(F0)(respectivelyLmax2L(F0)),fromLemma2.3.4itfollowsthatL2L(F0).Hence,L(F)L(F0).Similarly,L(F0)L(F);therefore,L(F)=L(F0). Proof.LetFbegiveninminimalnormalformandletFbeaforbidderinF.IfF=fwg,thenthelemmaholds.Assume,jFj2andletw2F.LetF0=(FnfFg)[fF0g,whereF0=Fnfwg.IfforallL2M(F)thewordw2sub

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FimpliesLcon F0.So,M(F)L(F0)andbyLemma2.3.6(i),L(F)L(F0).Hence,FF0,whichcontradictstheminimalityofF.Therefore,thereexistsalanguageL2M(F),suchthatw62sub Fthereisav2Fsuchthatv62sub F0.Therefore,M(F)L(F0)andagainbyLemma2.3.6(i)L(F)L(F0).ThisimpliesthatFF0contradictingtheminimalityofF.Hence,thereisL2Lsuchthat(Fnfwg)sub ThefollowingexampleshowsthatthereisaF,withF;H2Fandw2FforwhichLw6=Lvforeveryv2H.Example2.3.8 Intheaboveexample,thereisawordinoneforbidder(aa)whichisinthesubwordsofwordsfromanotherforbidder(baaandaaa).Thefollowinglemmageneralizesthisexample.Lemma2.3.9 Proof.LetL2M(F)andletF2Fsuchthatthereisw;v2Fwithw6=vandfw;vg\sub

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F.LetH2FandH6=F.IfHsub Proof.LetFbegivenasintheconditionsofthelemma.LetL2M(F)andletF2F.Then,thereisaw2Fsuchthatw62sub ItmaybethecasethatW(F)isnotsubwordfreeandthesubwordsofeverylanguagefromM(F)containallbutonewordfromeachforbidder.ThefollowingexampleshowsthattheconverseofLemma2.3.12doesnothold.Example2.3.13

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GivenA,anitelybranchinginnitetreeTArootedatisassociatedwithA.Figure2.1depictsthetreeTAforA=fa;bg. SinceeveryvertexhasexactlyjAjchildren,thetreeisnitelybranching.Becauseforeveryvertexvthereisachildvafora2A,thetreeisinnite.Also,notethatforeveryvertexvthereisauniquepathfromtov.Infact,thefollowingcorrespondenceholds.17

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Forexample,forA=fa;bgthepathassociatedwithw=aabais;a,aa;aab,aaba.Sinceeveryvertexinthetreehasauniquepredecessor,thereisauniquepathfromaabato. ThetreeTAcanbeusedtoobtainthetreesforthemaximallanguagesinanf-family.Denition2.3.16 ForatreeT,denotethelanguagedenedbyitsverticesbyLT.AlllanguagesL,suchthatthereisatreeT2TFwithL=LTaredenotedbyL(TF).ObservethatLTcon Notethatifw2FforsomeF2FissuchthatitisremovedfromTalongwithallwordsthatcontainwasasubword,thentherestofv2F(ifany)arenotnecessarilyinLT.Forexample,ifw;v2FforsomeF2Fwithw6=vandthereisaH2FwithH6=Fsuchthatv2H,thentherewillbetreeinwhichbothwandvwillberemoved. ConsiderFigure2.2,whichillustratesthetreesforthefourmaximallanguagesinM(F)fromExample2.3.2obtainedfromthebranchcuttingprocedure.18

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Proof.LetL2M(F).Then,fromLemma2.3.12itfollowsthatforeveryF2Fthereisaw2Fsuchthatw62sub 19

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LetFbeaforbiddingsetinminimalnormalform.WitheverylanguageL2M(F)atreerootedatisassociated.AnodeuatthetreeofLhasachilduafora2Aifandonlyifua2L.Clearly,thetreesmightbeinnite,buteachnodehasatmostcardinalityofthealphabetnumberofchildren.DenotethetreeforLwithTL. Anequivalent(extended)forbiddingsettoFisconstructedinthefollowingway.Anarbitrarysymbolfromthealphabetsetisdenotedwitha.Letw=w0abeawordsuchthatw0isanodeinTLforsomeL2M(F),butwisnotanodeinanyTL,L2M(F).ThenwisforbiddenbyFineverylanguageofL(F),i.e.,wisstrictlyforbiddeninL(F).SetGw=fwganddeneC=fGwjwisstrictlyforbiddeninL(F)g: NowconsiderawordvwhichisanodeinTL,butvaisnot.Inaddition,vaisanodeofsomeothertreeTL0.Then,theremustbeanodeuinTL0suchthatua0isnotanodeinTL0,butitisanodeinTL.Otherwise,LL0andLwouldnotbemaximal.Allwordslikevaandua0arecallednon-strictlyforbiddenforL(F).ConsiderP0=fvajvaisnon-strictlyforbiddenforL(F)g:

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Proof.LetL2M(F).LetF2^F.Byconstructionof^F,FiseitherinGorinQwforsomew2P.IfF2G,thenF=fwgforsomewandwisstrictlyforbidden.Hence,F6sub Fortheconverse,notethatbyconstructioneachforbidderfromFisin^F.IfF=fwgthenF2GsinceFisinminimalnormalform.IfFhasmorethanoneword,thenbyLemma2.3.7,thereisamaximallanguageLF2M(F)suchthatLFcontainsallwordsfromFbutone.Letw2Fbesuchthatw62LF.SinceQwcontainsallminimalsetsthatcontainwandarenotinanymaximallanguage,itfollowsthatF2Qw.Hence,L(^F)L(F). Theextendedforbiddingset^Fobtainedwiththeconstructionaboveiscalledmaximalsetofforbidders.TheexamplebelowusestheforbiddingsetfromExample2.3.2toillustratetheaboveconstruction.Example2.4.2

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Thecharacterizationofextendedf-familiespresentedbelowfollowsfromtheaboveconstruction.Theorem2.4.3 Proof.IfLsatises(i),thenthemaximalsetofforbiddersprovidesanextendedforbiddingset^FsuchthatL=L(^F).ByRemark2.3.5,themaximallanguagesofanextendedf-familysatisfy(i). Lemmas2.3.4and2.3.6showthatifLisanf-family,thenTheorem2.4.3(i)alsoholds.However,thenextexampleshowsthatTheorem2.4.3(i)maydenean22

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Theauthorsin[8]takethesmallestE-extensionsstepsinordertogeneratealanguagethatsatisesagivenenforcingsetE.ItfollowsthateverylanguageLthatcontainsXfromanenforcer(X;Y)2EhastocontainaminimalsetofwordsdenedbytheenforcingsetE.Inthissectiontheclassesoflanguagesdenedratherthanderivedbyanfe-systemareconsidered.Anewdenition,namelyofgeneratedandminimalgeneratedlanguages,isintroduced.Thesenotionsmaybeseenas\fastersteps"throughthe-treedenedin[8,9,40]andinthatsenseexpandthenotionofE-extensions.Denition2.5.1

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Ageneratedlanguagegm(X)iscalledminimal,ifnopropersubsetofitisageneratedlanguage. LetEbeanenforcingsetandletX2E(1).DenotethefamilyofgeneratedlanguagesofXwithrespecttoEbyGEXorsimplyGXwhenEisunderstood.ThefamilyofminimalgeneratedlanguagesofXwithrespecttoEisdenotedbyMEXorsimplyMXwhenEisunderstood.ThesetM(E)=[X2E(1)MXiscalledtheminimalgeneratedsetofE.Remark2.5.2 NotethatageneratedlanguagealwayssatisesE,whereasanE-extensionmaynot.ThefollowingexampleshowshowE-extensionsandminimalgeneratedlan-guagesdier.Example2.5.3 Lemma11.14in[40]showsaredundancyintheenforcingset:if(X;Y);(X0;Y0)aretwodierentenforcersinEsuchthatXX0andYY0,thenEE0whereE0=Enf(X0;Y0)g.Althoughnotassimple,thefollowingdenitionextendsthenotionofredundancy.24

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Inparticular,ifXX0andYY0,theneverygm(X)2ME0Xissuchthatgm(X)\Y6=;,hencegm(X)\Y06=;,i.e.,(X0;Y0)isredundant.Example2.5.5 Thefollowinglemmashowsthatredundantenforcerscanbeerasedfromtheenforcingset.Lemma2.5.6 Proof.ItisclearthatL(E)L(E0).LetL2L(E0).IfX06L,thenL2L(E).AssumeX0L.Since(X0;Y0)isredundant,thereisanenforcer(X;Y)2EsuchthatXX0andgm(X)\Y06=;forallgm(X)inME0X.SinceLcontainsatleastonegm(X)fromME0X,Lsat

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Thefollowingtwoexamplesshowthataninnitenitaryenforcingsetmayhaveaniteminimalgeneratedset,whichcontainsaninniteminimalgeneratedlanguage.Example2.5.11

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ThefollowinglemmashowsthataninnitenitaryenforcingsetwithniteM(E)musthaveaninnitegeneratedlanguage.Lemma2.5.13 Proof.SinceM(E)isnite,thereareanitenumberoffamiliesofminimalgeneratedlanguagesMX.DenotethesefamiliesbyM1;M2;:::;Mk,i.e.,M(E)=Ski=1Mi.SincethereareinnitelymanydistinctX's(duetoEbeinginnite)andnitelymanyMi's,theremustexistatleastoneMjsuchthatforinnitelymanyX'sinE(1),wehaveMX=Mj.Letgm(X)2Mj.Sincegm(X)isaminimalgeneratedlanguageforinnitelymanyX's,itfollowsthatgm(X)containsalltheseX'sassubsets.Hence,gm(X)isinnite.(Infact,allgeneratedsetsinMjareinnite.) Lemma2.5.13inusedtoshowthatinniteenforcingsetsdenenon-openfamiliesoflanguages(Proposition3.4.7).27

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Inthissection,thespaceP(A)withthemetricdenedin[10]and[40]isinves-tigated.Thismetriccomesnaturallyfromtheonedenedforthe!-wordsin[11]and[39]andtheoneusedinsymbolicdynamics(see[26,27]).Althoughthemetricisnatural,thestudyofthespaceofformallanguages(thelanguagespace)asatopological(metric)spacedidnotappearinliteratureuntil[16].Othertopologiesonformallanguagesareconsideredin[25]. DenotethesymmetricdierenceofL1andL2byL14L2.Denition3.1.1 (LanguageMetric)ThedistancebetweenanytwolanguagesL1andL2inP(A)is:d(L1;L2)=8<:1 2jforj=min

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Itiseasytoseethatddenedaboveisametric.TheopenballcenteredatLwithradiusisthesetofalllanguagesthatareatadistancelessthanfromL.ItisdenotedbyBd(L;).Clearly,K2Bd(L;)ifandonlyifK6m=L6mforanymsuchthat2m<.Thereisacloserelationshipbetweenthelanguagemetricandtheonedenedfor!-wordsin[11],whosedenitionisrecalledbelow.Denition3.1.2 (!-wordMetric)The!-worddistancebetweenanytwowordsandinA!is:(;)=8<:1 2j;forj=min NotethatKandLbelongtothesamecylindersetwithboundmifandonlyifL\A6m=K\A6m.ThecollectionofcylindersetscorrespondstotheopenballsforP(A)andhenceisabasisforthetopologydenedbyd.Givenm,P=fC(K)mjK2P(A)gformsanitepartitiononP(A).Forexample,ifm=1andA=fa;bg,thenthereareeightcylindersetsinthepartitionP.Namely,P=f;;fg;fag;f;ag;fbg;f;bg;fa;bg;f;a;bgg.Forexample,thecylindersetC(;)1consistsofalllanguageswhosewordshavelengthgreaterorequalto2.ItiseasytoseethateverylanguagefromP(A)belongstoexactlyoneoftheseeightcylindersets. TheabovedenitioncorrespondstothedenitionforcylindersetsinX!denedwithCi(a0ak)=fjii+1k=a0a1akg(seeforex.[26]).29

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Proof.Obviously,isabijection.LetBd(L;)with>1 2mbeanopenballinP(A).Foreachw2L6mleti(w)betheorderofwinA,i.e.,i(w)=1(w).Let(L)=andletj=max Thisprovesthefollowingtheorem.Theorem3.1.5 Presentbelowisadirectproofofthetheorem.Atopologicalspaceiscalledperfect,ifithasnoisolatedpoints.Notethatnoisolatedpointisalimitpoint(see[29]).Theorem3.1.6 Therefore,itishomeomorphictotheCantorspace. Proof.Compact.LetfLngn0beasequenceoflanguagesinP(A).ItissucienttoshowthatfLngn0containsaconvergentsubsequence.IffLngn0isanitefamily30

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31

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Proof.ConsideracylinderC(K)mcenteredatK2P(B).LetL=fwjwisinalanguageinh1(K6m)g.ThecylindersetC(L)mcenteredatLmapsintoC(K)m.LetL6m1=L6mandh(L1)=K1.Letu2K1andjujm.Thenthereisawordu02L1withh(u0)=u.SinceL6m1=L6m,wehaveu02Landhenceu=h(u0)2K,i.e,K6m1K6m.BythesymmetryoftheargumentforLandL1wehaveK6m1=K6m. Conversely,assumethatthereisa2Awithh(a)=.Theneitherh(A)=fgorthereisbsuchthath(b)6=.ConsiderL0AbAbAwherebissuchthath(b)6=.Letm>jh(b)j.ForeveryndeneLn=L0[fakbjkng.ThenL6n0=L6nnforallnbutd(h(L0);h(Ln))=2jh(b)j>2m.Hence,hisnotcontinuous.Intheeventthath(A)=fg,itisobviousthathisnotcontinuoussinceh(P(A))=f;;fgg. ByProposition3.1.4,P(A)andf0;1g!arehomeomorphic.Theorem2.1in[39]classiesthecontinuousmapsonX!.Itstatesthatamap':X!!Y!iscontinuousifandonlyifitisanextensionofatotallyunbounded(innitelanguagesmapintoinnitelanguages)andsequential(imageofaprexofawordisaprexoftheimageofthesameword)mapping':X!Y.LetX=Y=f0;1gandconsidertwoarbitraryalphabetsAandB.BothP(A)aswellasP(B)arehomeomorphictoX!.Ahomomorphismh:P(A)!P(B)extendstoamap^h32

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Anattempttodescribethesamefamilyusinginnitesequenceswouldbequiteburdensomesinceonehastohaveathandtheorder(index)ofallwordsthatcontainabasasubword.ThenconvertthelanguageK=fwjab2sub Innitesequenceswillnotbediscussedfurtherinthiswork.Proposition3.2.2 Proof.Supposehdoesnotextendtoacontinuoush.Thenh(a)=forsomea2Aandh()=.Contradiction,sincehisone-to-one. 33

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LetFIN,REG,CF,CS,REdenotethefamiliesofnite,regular,contextfree,contextsensitiveandrecursivelyenumerablelanguagesrespectively.Thissectionshowsthatthesefamiliesdonotcorrespondto\nice"topologicalspaces. AsequenceoflanguagesfLngn0isconvergenttoalanguageL,ifforeachm2NthereisM2N,suchthatL6mi=L6mwheneveri>M.ThisisdenotedbyLn!L. Thenextlemmafollowsdirectlyfromthedenitions.(Seealso[10,40].)Lemma3.3.1

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Proof.(i) FollowsfromthefactthatifKisanylanguageandK62L,thenK6n!KandeachK6nisnite.So,KbelongstotheclosureofL,i.e.,closure(L)6=L.(ii) Followsfrom(i).(iii) Followsfromthefactthateverylanguageisalimitofasequenceofnitelanguages(Lemma3.3.1).(iv) LetRbeanonr.e.languageanddeneRj=RnR6j.ForeverylanguageL2P(A)andforeveryj>0wehavethatd(L;L6j[Rj)<2j. TheabovetheoremshowsthatthewellknownChomskyfamiliesoflanguagesdonothave\nice"propertiesinthistopology.Inthestudyofformallanguagesthesefamiliescontainlanguagesthatareclassiedbymeansmuchdierentthantopo-logicalpropertiesandareseparatedeitherbythetypesofautomatathatrecognizethemorbythetypesofgrammars.Inthissense,thetworegularlanguagesa+andawouldbeconsideredveryclosetoeachother.Butintopologicalsense,theyareatdistance1fromeachother(thelargestdistancepossible!).OthertopologiesonthespaceofformallanguageshavealsoshowntobenotsuitableforcharacterizingandclassifyingtheChomskyhierarchy[25].3.4TopologicalPropertiesoffe-Families

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Proof.SinceFisnotempty,thereexistsaforbidderFinFsuchthatFisnite,non-empty,andthewordsinFaredistinctfromtheemptyword. LetL2L(F)begivenandchoosea2A.Thenforeachs>0considerLs=L[faswjw2Fg.ObservethatLs2Bd(L;1 2s)butLs62L(F),sinceFsub(Ls).Hence,L(F)isnotopen. NotethatifFistrivial,thenbyRemark2.1.3,L(F)=P(A)andisopenbydenition.Also,theaboveproofshowsastrongerresult,whichwestateinthefollowingcorollary.Corollary3.4.3 Proof.Proceedasinthepreviousproof,exceptthatL2V.SinceLs62L(F)foreachs,thenLs62V.So,Visnotopen. NotethatifV=;,itisopenbythedenition. Authorsin[8]and[40]showthatL(E)areclosedsetsinP(A).Hencethesesetsarealsocompact.Thefollowingdiscussesunderwhatconditionstheyareopen.Proposition3.4.4

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2m)L(E).Hence,L(E)isopen. Proposition3.4.4conrmsthattheboundaryconditionsofniteenforcingsetsarenotveryrestrictive,andbytheobservationinTheorem3.3.2theycontainnonr.e.languages.Infact,wehavethefollowingobservation.Remark3.4.5 Theabovepropositionshowsthatafe-systemwithemptyforbiddersandniteenforcersisanopensetthatcontainsbasiselementsforthetopologyonP(A).Theinniteenforcingsetshavepotentialtoprovidefamiliesoflanguagesthatdonotcontainnonr.e.languages.Thefollowingobservationsshowthatinthecaseofinniteenforcingsets,thedenedfamilyisalwaysnon-open.Example3.4.6 Theenforcingsetdiscussedintheaboveexampleisnotnitary,butasimi-larargumentcanbemadeforitsnitaryequivalentenforcingsetE0=f(;;fw1g),(fw1g;fw2g),:::gandtheinnitelanguageL=fw1;w2;:::g.Thefollowingpropo-sitionshowsthattheaboveexampleispartofageneralrule.Proposition3.4.7

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2m. InthecasethatM(E)isniteitfollowsfromLemma2.5.13thatthereexistsaninniteminimalgeneratedset,whichwedenotebyK.Asinthecaseofinnitenumberofgeneratedsets,thefactthateveryopenballcenteredatKcontainsalanguagethatisnotinL(E)isshown.SinceKisinnite,foreverym0thereisawordwm2K,suchthatjwmj>m.Nowforeverym0constructthelanguageLm=Knfwmg.Thenforeverym0,Lm2Bd(K;1 2m)butLm62L(E)becauseLmisapropersubsetofaminimalgeneratedset. Propositions3.4.4and3.4.7establishatopologicaldierencebetweenniteandinniteenforcingsets.Theyshowthatinnitenitaryenforcingsetscannotbeequivalenttoniteenforcingsets,becausethersttypeofsetsdescribesnon-openfamiliesoflanguagesandthelatter-open.Thisfactisstatedinthefollowingcorollary.Corollary3.4.8

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NotethatinthecasethatL(F;E)=;thefe-familyisopen,butasExamples2.2.2and2.2.3showFmaybenonemptyandEmaybeinnite. Althougheverycylindersetisincludedinane-family(Remark3.4.5),therearecylindersets(openballs)thatdenefamiliesoflanguagesthatcannotbedenedbyfe-systems.Example3.4.10 Theaboveexampleextendstothefollowingfact.Proposition3.4.11 Proof.LetPandLbeasdenedintheproposition.Supposethereexistsafe-systemsuchthatL()=L.ThenthelanguageK=AnPbelongstoL().LetthemaximumlengthofawordinPben.ThenallwordsoflengthgreaterthannareinK,whichimpliesthatF=;.ButthenthewordsfromPcannotbeexcludedbyenforcingonly.Contradiction,hencenosuchexists. Proposition3.4.11canbeprovedbytopologicalobservation,aswell.LetthemaximallengthofwordsinPben.LetL2LandconsiderthecylinderC(L)mforsomesomemn.Then,C(L)mL,henceLisopen.SinceL6=;,fromTheorem39

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Proof.LetP=Kc\A6m.ThenthecorollaryfollowsfromProposition3.4.11. Proof.BylettingF=;andenforcingA6masinRemark3.4.5weobtainafe-familyequaltothecylinderset.TheconversefollowsfromCorollary3.4.12. 40

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Proof.LetL2L(h(F))andletK=h1(L).Sincehisonto,h(K)=L.Sincehisamorphism,ifw2sub 41

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Proof.LetL2h(L(F)).ThenthereisK2L(F),suchthath(K)=L.LetF02h(F),thenthereisF2F,suchthath(F)=F0.SinceK2L(F),Kcon F.Thismeansthatthereisw2F,suchthatw62sub F0byshowingthatw062sub F0,whichimpliesL2L(h(F)). ThefollowingexampleshowsthatthatsurjectivityisessentialinProposition4.1.1,injectivityisessentialinProposition4.1.2,andequalitydoesnotnecessarilyholdinbothpropositions.Example4.1.3 LetA=fa;b;cgandB=fd;egwithh(a)=h(b)=dandh(c)=e.LetFcontainonlyoneforbidderF=fac;bcgandletK=fbcg.Thenh(K)=h(F)anditisnotinh(L(F)),buth(K)62L(h(F)).ThisexampleshowsthatequalitydoesnotalwaysholdinProposition4.1.1.It,also,showsthatinjectivityisessentialinProposition4.1.2.(b) ToobservethatProposition4.1.1doesnotholdifhisnotsurjectiveconsiderA=fa;bgandB=fc;d;egwithh(a)=candh(b)=d.LetF=ffaagg.ThenL=feg2L(h(F)),butL62h(L(F))sinceh(L(F))fc;dg.This42

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Similarpropertiesholdforenforcingfamilies.Inthiscasetherequirementthathisamorphismisnotnecessary.Proposition4.1.4 Proof.LetL2L(h(E))andletK=fwjh(w)2Lg.Sincehisonto,h(K)=L.SupposeKnsat Proof.LetL2h(L(E)).ThenthereisK2L(E),suchthath(K)=L.Let(X0;Y0)2h(E)withh((X;Y))=(X0;Y0).IfX06L,thenLsatisesthisenforcertrivially.IfX0L,thenXKsincehisinjective.SinceKsat ConsideragainA=fa;b;cgandB=fd;egwithh(a)=h(b)=dandh(c)=e.LetE=f(fabg;fccg)g.SincethelanguageK=faagmapsintoL=fddg,wehavethatL2h(L(E)).However,L62L(h(E)).Thisexampleshowsthatequalitydoes43

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Thenextcorollaryfollowsstraightforward.Corollary4.1.6

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Notethateventhoughtheimageofanf-familyundersuchmorphismsmightnotbeanf-family,itcanstillbeanfe-family,asshowninthenextexample.Example4.2.2 Theaboveproposition,also,followsfromLemma2.3.4since,whenjh(a)j>1forsomesymbola,wecanndanFsuchthatthemaximallanguagesinM(F)mapintolanguagesthatarenotfactorial.Ifthereexistsamorphismh:P(A)!P(B)thatmapseveryf-familyintoanf-family,thenh(A)B.Thefollowingexamplepresentsamorphismofthistypemappingf-familiesintof-families.Example4.2.3 ConsiderA=fa;bgandB=fc;dgandamorphismhsuchthath(a)=h(b)=c.IfF=ffaa;abg;fbag;fbbggthenL(F)consistsofalllanguagesthataresubsetsofa[b.Hence,h(L(F))isthefamilyofalllanguagesthataresubsetsofcandF0=ffdgg.(b) Forh,A,andBasabove,setF=ffab;bag;faag;fbbgg.ThenL(F)consistsoflanguagesthatdon'thavewordsoflengthlargerorequalto3.Thus,wecansetF0=ffdg;fcccgg.45

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Proof.LetFbeaforbiddingset.LetLbeamaximallanguageinh(L(F)).Then,thereisalanguageL02M(F),suchthath(L0)=L.Itissucienttoobservethat(i)fromTheorem2.4.3holdsforL.LetKL.TheneverywordinKhasapreimageinL0.SothereisK0L0suchthath(K0)=K.SinceK02L(F)itholdsthatK2h(L(F)).Observethat,sincehmapssymboltosymbol,Lisfactorial.Considerw2Landx2sub Thefollowingresultstatesthatsurjectivityisessentialformappinge-familiesintoe-families.Proposition4.2.5 Proof.Thepropositionfollowsfromthefactthatthereexistsawordw2Bsuchthath1(w)=;.SupposethereexistsE0suchthath(L(E))=L(E0).LetL2h(L(E))andconsideralanguageK2L(E0)thatcontainsL[fwgasasubset.(Suchalanguagealwaysexists.Inparticular,Bisonesuchlanguage.)ThenKisinh(L(E)),aswell,whichcontradictsthefactthath1(w)=;.Hence,nosuchE0exists. Althoughtheimageofane-familyunderanonsurjectivemorphismisnotane-family,itcouldbeanfe-family,asshowninthefollowingexample.Example4.2.6

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LetA=fa;bgandB=fcg.Leth:A!Bbeamorphismsuchthath(a)=h(b)=c.ConsiderE=faag;fa3g;faag;fa4g;fabg;fa5g;fbag;fa5g;fbbg;fa5g: ConsiderA=fa;a0;b;b0;c;c0gandB=fa;b;cg.Leth:A!Bsuchthath(a)=h(a0)=a,h(b)=h(b0)=b,andh(c)=h(c0)=c.LetE=f(fa;bg;fabcg);(fa0;b0g;fabcg);(fa0;cg;fabcg);(fa0;c0g;fabcg);(fb0;cg;fabcg);(fb0;c0g;fabcg)g: Recallthatanopenmapisafunctionthatmapsopensetsintoopensets.47

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Proof.Assumethateveryfe-familymapswithhontoanfe-family.Thensuchafamilywithemptyenforcers,orwithanemptyforbiddingsetalsomapsintoanfe-family.Let=(F;E)besuchthatF=;andEbenite.ThenL()isopenandh(L())isopen,aswell.Thismeansthatin0=(F0;E0)theforbiddingsetmustbeemptyandthesetofenforcersmustbenite.ByProposition4.2.5hissurjective. Unfortunately,theconversedoesnotholdevenwhenhmapssymbolstosymbolssurjectively.Considerthefollowingexample.Example4.2.9

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AnisomorphismbetweentwosimplegraphsGandHisavertexbijection':VG!VHsuchthatforeachu;v2VG,uandvareadjacentinGifandonlyif'(u)and'(v)areadjacentinH.Implicitly,thereisalsoanedgebijectionEG!EHsuchthatuv!'(u)'(v).TwosimplegraphsGandHarecalledisomorphicifthereisanisomorphismfromGtoH. Atrivialgraphisagraphconsistingofonevertexandnoedgesandanullgraph49

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AsubgraphofagraphGisagraphHwhosevertexandedgesetsaresubsetsoftheseofG.Forthatmatter,anygraphisomorphictoHis,also,consideredasubgraphofG.ThisisdenotedbyHGorH
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GivenanitesetofgraphsofnontrivialgraphsF=fH1;:::;HngdenethegraphSH1Pi1H2:::Pin1HnsuchthatH1isconnectedthroughapathPi1toH2andcontinuingthiswayHn1isconnectedthroughPin1toHn.Furthermore,V(H1)\V(Pi1)=fu01g,V(Pi1)\V(H2)=fu2g,:::,V(Pin1)\V(Hn)=fung,whereallu'saredistinctandeverygraphfromFandeverypathisotherwisevertex-disjointfromeveryothergraphorpath.(Figure5.1depictsthegraphSC3P4C4.)5.2ConnectingGraphs OnesuchGcanbeobtainedbyorderingthegraphsinF=fH1;:::;HngandconnectingHiwithHi+1withanedgethathasonevertexinHiandtheothervertexinHi+1.Obviously,GisconnectedandFsub

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AnotherconnectinggraphG0ofFcanbeobtainedbytakingthegraphwiththemaximumnumberofvertices(saym)amongthegraphsinFandlettingG=Km.Then,byremovingedgesuntilthegraphisnolongerconnectedornolongeraconnectinggraphofFaminimalconnectinggraphSG0isobtained. Anitesetofgraphsmayhavemanyminimalconnectinggraphs.Considerthefollowingexample.Example5.2.3 AnotherwaytodeneminimalityistorequirethatSbevertex-minimal.Inthiscase,D4istheuniqueminimalconnectinggraphfortheFconsideredinExample5.2.3.Denition5.2.4 Thefollowingexampleshowsthatevenifthisdenitionofminimalityisused,theminimalconnectinggraphisnotnecessarilyunique.52

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Fortherestofthischapter\minimal"connectinggraphsareasinDenition5.2.1. Itisobviousthatifagraphisaconnectinggraphofanitesetofgraphs,itisalsoaconnectinggraphofeverysubsetofthissetofgraphs.Thisfactisstatedformallyinthenextproposition.Proposition5.2.6 Thefollowingpropositiongeneralizestheaboveexample.Proposition5.2.8 Proof.LetS2Cmin(F).ByProposition5.2.6S2C(F0).Hence,thereisaT2Cmin(F0)suchthatTS.SinceT2C(F),itfollowsthatT=S,henceCmin(F)Cmin(F0).Conversely,letS2Cmin(F0).SinceHS,itfollowsthatS2C(F).Then,thereisT2Cmin(F)suchthatTS.ByProposition5.2.6T2C(F0)anditfollowsthatT=S.Hence,Cmin(F0)Cmin(F). 53

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Proof.LetFbeaconnectingfreenitesetofgraphs.LetH;K2F.SinceFisconnectingfree,itfollowsthatthereisaconnectinggraphS2C(FnfHg)suchthatHS,whichimpliesthatHK.Similarly,KH.Hence,Fissubgraphfree.Example5.2.11showsthattheconversedoesnothold. Proof.LetFbeconnectingfreeandF0F.LetK2F0.KSforeveryS2C(F0nfKg)impliesKSforeveryS2C(FnfKg).Hence,F0isconnectingfree. ItiseasytoseethatifFhasonlytwographsthenthenotionsofconnectingfreeandsubgraphfreecoincide.Proposition5.2.12andRemark5.2.9provethefollowingremark.54

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Thefactbelowcaneasilybeprovedanditisusedinthepropositionthatfollows.Remark5.2.16 Clearly,ifFisasintheaboveremarkandn3,thenCmin(F)isasingleton.Proposition5.2.17 Proof.LetnandHbegiven.IfFisnotsubgraphfree,thenCmin(F)isasingleton.AssumeFissubgraphfree,i.e.,PnHandHPn.ByRemark5.2.16n4.Also,Hisnotapath.ThelongestpathinHcanbe\extended"toPn.TherearenitelymanyverticesandpathsinHandnitelymanywaysofextendinganypathfromanyvertextoPn.Hence,Cmin(F)isnite. Proof.LetF;H1;H2,andH3areasintheconditionsofthepropositionandletSn=SH1PnH2wheren2.IfthereisaksuchthatforeverynkitholdsthatH3Sn,thepropositionholds.Otherwise,foreverykthereisanksuchthatH3Sn.Considerthesmallestsuchn.Then,ifforeverymn,H3Sm,thepropositionholds.Otherwise,thereisamsuchthatm>nandH3Sm.IfforeverylmH3Sl,thepropositionholds.Otherwise,thereisal>msuchthatH3Sl,butinthiscaseforeveryplitholdsthatH3Sp.Toseethissupposethereisap>lsuchthatH3Sp.Then,H3H1;H2,sinceotherwiseH3Sl.Also,H3SH1PpandH3SPpH2forotherwiseH3Sl.(HereSH1PpisH1withapathPpthatsharesacommonvertexwithH1andisotherwisedisjointfromH1andSPpH2isdenedanalogously.)Hence,H3is\partly"inH1,\contains"theentirePp55

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AsanexamplethatillustratestheaboveproofconsiderF=fC3;C4;Tg,whereTisagraphconsistingofaP3witheachendbeingofdegree3suchthatthetwoneighborsoftheendvertexnotonthepathareofdegree1.Then,k=n=2,m=3,andl=4.Denition5.2.19 ThesetFfromExample5.2.20issubgraphfree,butnottailedsubgraphfree.Thefollowingexampleillustratestheconverse.Example5.2.21

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TherearenitesetsofgraphsFnotcontainingpathsthathaveaniteCmin(F).Proposition5.2.22 Proof.Letj2f1;2;3gbegivenandconsiderHj.LetSHiPmHlbesuchthati6=l,j6=i;l,andi;l=1;2;3.IfthereisassuchthatforeverymsHjSHiPmHl,thenthepropositionholds.Otherwise,byProposition5.2.18thereisaksuchthatforeverynkHjSHiPnHl.IfHjtHi;Hl,thenHjmustbe\partly"inHi,\partly"inHlandmustcontaintheentirePn.SincePnHjforallnkitfollowsthatHjisinnite,whichcontradictsthefactthatH3isnite.Therefore,HjisatailedsubgraphofatleastoneofthegraphsinFnfHjg. Proof.ConsiderSn=SH1PnH2.SinceFisbothsubgraphandtailedsubgraphfree,thecasewhereSnisnotgoingtobeminimaliswhenoneofthegraphsis\partly"inH1,\partly"inH2andcontainstheentireSnforsomen.SincebothH1andH2arenitegraphs,thereisaminimalksuchthatforeverynkSnisminimal.

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Proof.ItisclearthatFdoesnotcontainanypaths.LetF=fH1;H2;:::;Hng.Ifn=2,byProposition5.2.23,Cmin(F)isinnite.Letn>2.ConsiderthegraphS=SH1Pi1H2:::Hn1Pin1Hnwhereij2foreveryj=1;:::;n1andthevaluesforijwherej=1;:::;n1aredeterminedasfollows.Obviously,thegraphSconnectsFforanyij.ConstructSinthefollowingway.LetS=SH1PlH2.ByProposition5.2.23,thereisasmallestksuchthatSisminimalforeverynk.Letl=k.ByProposition5.2.22,thereisssuchthatforeverymsH3SH1PmH2.Lettbethesmallestsuchsandleti1=maxfk;tg.LetS=SH1Pi1H2PlH3.IfH1Sl=SH2PlH3forsomel2,thereisasmallestksuchthatforeverymkH1Sl.Letl=k.Otherwise,letS=SH1Pi1H2Pi2H3wherei2=landmovetoH4(ifany).CheckwhetherH4SH1Pi1H2andwhetherH4SH2Pi2H3andincreasei1andi2sothatthisnolongerholds.Then,\add"H4toS.ContinuethiswayandateverystepthatHiisaddedcheckwhetherH1;:::;Hi2aresubgraphsofSl=SHi1PlHiforsomel,andinsuchcaseexpandPi1untiltheyarenolongersubgraphsofSl.Nowthatallvaluesforijwherej=1;:::;n1aredetermined,eachtimeaconnectingpathinSisincreased,thiswillresultinanewminimalconnectinggraphofF.58

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Proof.IfFhasnopaths,thetheoremholdsbyProposition5.2.26.IfthereisapathPk2F,(thenn3)letPk=Hn.NotethatFcanhaveatmostonepath.ProceedasintheproofofProposition5.2.26untilgraphHn1is\added"toS.IfHnisalreadyasubgraphofS,thentheproofiscompleted.IfHnisnotalreadyasubgraphofSthenPin1HnisapathPlwithsuchalengththatPkS,butPk+1S.Thetheoremnowfollows. Proof.IfjFj=1,thenCmin(F)isnite.SupposethatjFj2suchthatforeveryH;K2Feitheroneofthemisasubgraphoftheotheroroneofthemisatailedsubgraphoftheother,i.e.,HKorKHorHtKorKtH.LetF=fH1;:::;Hng.Then,eitherH1H2orH2H1orH1tH2orH2tH1.InthersttwocasestakethelargergraphandcallitH0,inthelasttwocasesextendoneofthegraphsbyapathtoobtainaminimalconnectinggraphofH1andH2.Thiscanbedoneinnitelymanyways.LetH0beonesuchgraph.ProceedtoH3.Then,foreachofHi,wherei=1;2eitherHiH3orH3HiorH3tHiorHitH3.Ineachcase,theminimalconnectinggraphsofH1andH2canbeextendedinnitelymanywaystoobtaintheminimalconnectinggraphsofH1;H2,andH3.ContinuethiswayuntilHn.Consequently,Cmin(F)isnite.Thepropositionnowfollows. 59

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Figure5.5depictstherstfourlevelsofthegraphGU.Eachlevelcorrespondstothenumberofedgesinitsgraph.Notethatthereisonegraphateachofthelevels0,1,and2.Thereare3graphsatlevel3,5graphsatlevel4,12graphsatlevel5,andsoon.Allpathsthatstartatanyvertexareinnite. Thefactsinthenextpropositionfollowdirectlyfromthedenitionsandareeasytoprove.Proposition5.2.31(i)

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Foranon-connectedgraphX,theindexgisaddedinC(Xg)todistinguishbetweentheconnectinggraphsofthecomponentsandtheconnectinggraphsoftheset.Thus,ifXisviewedasasetofgraphs,theconnectinggraphswouldbeC(X). Asstatedinthenextsection,non-connectedgraphsXappearingraphfe-systemsonlyasarstcomponentofanenforcer.Inthiscase,nottheminimalityofconnectinggraphsofXperseisofinterest,butratheraminimalconnectinggraphofXthatcontainsaspeciccopyofXandisembeddedinaspeciedconnectedgraphG.ThistypeofminimalityisrelativetoGandtothecopyofXinG.Hence,thefollowingdenition.Denition5.2.33 Thus,N1;0;0isanextensionbyanedgeofK3.62

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LetXbeanitenon-connectedgraph.ThenXhasanitenumberofcompo-nentssayH1;:::;Hn,wheren2.Letthenumberofverticesofeachcomponentbekiandletm=ni=1ki.Then,thegraphKmisaconnectinggraphofX.Consideracopyof^XofXinG.ConsiderthesetofallgraphsTthatcanbeobtainedbyremovinganynumberofedgesfromG,suchthatTisconnectedandcontainsthecopy^X.Then,thereisaminimalconnectinggraphSof^XrelativetoG,suchthatS=TforsomeT.NotethatSisnotnecessarilyunique. NowconsideraninnitesequenceofextensionsbyanedgeofKm,K1m;K2m;:::suchthatKimKi+1m.Then,XKimforeveryi1.Hence,ifXisanitenon-connectedgraph,thenCG(X)isinnite. Insteadoftheextensionsbyanedge,onecanalsotakethesequenceKk,Kk+1,:::,Kn,:::,whichalsoshowsthatCG(X)isinnite. AnotherwaytoconstructaconnectinggraphforXistoordertheconnectedcomponentsandconnectthembypaths.Morespecically,letfH1;:::;HngbetheconnectedcomponentsofXinsomeorder.LetS=SH1Pi1H2:::Pn1Hn.Then,SisaconnectinggraphofXandvaryingthelengthofthe\connecting"pathswillproduceinnitelymanyconnectinggraphsofX.Inaddition,thereisaninnitesequenceofextensionsbyanedgeofSS1;S2;:::,suchthatSiSi+1andXSiforeveryi1.63

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F,ifGisconnectedandF6sub FforallF2F.Gncon ForaforbiddingsetFthefamilyofF-consistentgraphsisL(F)=fGjGcon ThefamilyL(F)issaidtobedenedbytheforbiddingsetF.AfamilyLiscalledaf-family,ifthereisaforbiddingsetFsuchthatL=L(F).Remark5.3.2 ForGconnectsF,i.e.,L(F)c=C(F). Thefollowingboundaryobservationsstatethatifnothingisforbiddeneverythingisallowedandthatthetrivialandnullgraphsarealwaysinaf-familyofgraphs.Remark5.3.3(i) Thenullgraph;andthetrivialgraphareinL(F)foreveryF. Ingeneral,forbiddersmaycontainmorethanoneelement.Thefollowingexam-pleshowshowageneralforbiddingsetdiersfromastrictone.64

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Historically,theconceptofforbiddengraphshasbeenusedtocharacterizeHamil-toniangraphsandinTurantypeproblems(seeforex.[2,7,17]).Foracomprehen-sivelistofreferencesreferto[2,18].Inexistingliterature,forbiddengraphsareanite(orinsomecasesinnite)setofgraphsfF1;F2;:::gwhereeachoftheseFiisaforbidden(induced)subgraphofG.Inthatrespect,ourforbiddingsetsdenitiondiersfromtheforbiddengraphsinthat,itemploysforbiddersthatarenotneces-sarilysingletonsbutanitesetofgraphsanditconsiderssubgraphsasopposedtoinducedsubgraphs.Denition5.3.7 AgraphGissaidtosatisfyanenforcer(X;Y)ifGisconnectedandwheneverXGthereisYi2YsuchthatX
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Toeasenotation,enforcersaresometimesdenotedwithE,i.e.,E=(X;Y).InthecasethatX6G,Gissaidtosatisfytheenforcertrivially.EnforcersinwhichX=;orX=arecalledbrute.Inthiscase,agraphfromYhastobeasubgraphofGinorderforGtosatisfytheenforcer. Anenforceriscalledtrivialifeverygraphsatisesitandnontrivialotherwise.Inlanguageenforcingsetsenforcers(X;Y)whereX\Y6=;arecalledtrivial,becauseeverylanguagesatisesthem.Thus,ifalanguageenforcingsetconsistsoftrivialenforcersonly,itdenestheentireP(A).Ifalanguageenforcingsetdoesn'thavetrivialenforcers,thenitdenestheentirelanguagesetifandonlyifitisempty. Thefollowingisaninvestigationofwhetherthegraphenforcingsetscanhavetrivialenforcers.Proposition5.3.8 Proof.Let(X;Y)beanenforcer.IfXisaconnectedgraph,thenXnsat Sincetherearenotrivialenforcersinthegraphcase,wehavethefollowingremark.Remark5.3.9(i) 2L(E)foreveryEthatdoesnothavebruteenforcers.66

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Insomesense,strictenforcers\force"thegraphfromYintothegraphGforeachoccasionofXinG.Considerthefollowingexampleconsistingofstrictenforcersonly.Example5.3.11 Thetwonotionsofforbiddingandenforcingongraphsarecombinedinthefollowingdenition.Denition5.3.12 AfamilyofgraphsLiscalledaforbidding-enforcingfamilyorfe-family,ifthereexistsafe-system(F;E),suchthatL=L(F;E).Example5.3.13 Twosetsofforbidders(ortwoenforcingsets,ortwoforbidding-enforcingsys-tems)areequivalent,iftheydenethesamefamilyofgraphs.Theequivalencerelationisdenotedby.Also,twoforbidders(enforcers)areequivalent(againdenotedby)ifthesingletonforbidding(enforcing)setscontainingeachofthemareequivalent,e.g.,FF0ifandonlyifFF0whereF=fFgandF0=fF0g.Similarly,EE0ifandonlyifEE0whereE=fEgandE0=fE0g.67

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Thenextpropositionstatessomeoftheimmediatepropertiesofgraphfe-systems.Theyfollowdirectlyfromthedenitionsaboveandmatchexactlythepropertiesoflanguagefe-systemsasstatedin[40].Proposition5.3.15 Theaboveexampleshowsthatthereareforbidders(forbiddingsets)whichcouldbereplacedentirelybyenforcingsets.Proposition5.4.2

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TheabovepropositioncanbeextendedtoageneralforbiddingsetFwithmorethanoneforbidderbyincludingenforcerslikeEFforeveryF2FandconsideringtheirunionE=[F2FEF.Then,L(F)=L(E).Thenexttheoremstatesthisconclusionformally.Theorem5.4.3 Proof.LetFbeaforbiddingset.IfF=;,thenletE=;.ByRemarks5.3.3and5.3.9L(F)=L(E)=U.LetFhaveatleastoneforbidder.ForeveryforbidderF2FconstructtheenforcingsetEFasinProposition5.4.2.ConsiderE=[F2FEF.LetGcon Thisdoesnot,however,renderforbiddingsetsobsolete.Itwouldbemuchmorepracticalandusefultorepresentagraphfamilybynitestructuresliketheforbid-dersF,ratherthaninnitesetslikeEF.Inaddition,fe-systemsmaypotentiallybeappliedinDNAcomputingandselfassemblyofgraphs,wherethenitenessofconstraintsisofgreatimportance.5.5CharacterizationsofSomeClassesofGraphsbyfe-Systems

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Proof.FollowsfromthefactthatL(F)containseverygraphthatdoesnothaveacycle. Figure5.8illustrateshowthepresenceofedgesaorbwillmakethegraphnonconsistentwithfC4gorfC3grespectively. TheBipartiteGraphCharacterizationTheoremstatesthatagraphisbipartiteifandonlyifthelengthofeachofitscyclesiseven(see[18]).Hencethefollowingf-familycharacterizationofbipartitegraphs.Proposition5.5.2 (Bipartitegraphs.)LetF=ffC3g;fC5g;:::;fC2k+1g;:::g.ThenL(F)containseverygraphthatdoesnothaveanoddcycle,i.e.,L(F)=fGjGisbipartiteg.

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(Pathsandcycles.)LetF=ffK1;3gg.ThenL(F)=fPnjn0g[fCnjn3g.Inotherwords,GisapathoracycleifandonlyifG2L(F). Proof.Clearly,L(F)containseveryconnectedgraphthatdoesnothaveavertexwithdegreemorethan2. Thenextcorollaryprovidesanf-familycharacterizationofpaths.Corollary5.5.4 (Paths.)LetF=ffK1;3g;fC3g;fC4g;:::g.Then,L(F)=fGjGisapathg. Inpropositions5.5.1,5.5.2,and5.5.3eachforbidderisasingleton.Thus,thegraphsappearingintheforbiddersareinsomesense\strictly"forbiddenassub-graphs.Thesepropositionsshowthatsomeclassesofgraphscanbeclassiedusingforbiddersonly.Thefollowingcharacterizationshowsthatasingletonenforcingsetdenestheclassofcompletegraphs.Proposition5.5.5 (Completegraphs.)LetE=f(P3;fC3g)g.ThenEdenestheclassofcompletegraphs,i.e.,L(E)=fGjGisacompletegraphg. Proof.Ifagraphiscomplete,anythreeverticesforma3-cycle.Ontheotherhand,supposeG2L(E)andGhastwoverticesuandvthatarenotadjacent.SinceGisconnected,thereisapathPnwithn3fromutov.Lettheorderoftheverticesinthepathbeu1=u;u2;:::;un1;un=v.Sinceu1u2u3isaP3,thentheedgefu1;u3gmustbeinthegraph.Similarly,u1u3u4impliesfu1;u4gisinthegraph.Continuingthisway,u1un1unimpliesthatfu1;ung2E(G). Thefollowingdenitionoftrimmedextensionsetsisusedinthecharacterizationofk-regulargraphs.71

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Anextension^HTiscalledtrimmed,ifitisobtainedfromsome^Hasfollows.If^Hissuchthatforeveryi=1;:::;kandeveryj=1;:::;kitholdsthatvi6=v0jwherevi;v0j2V(^H),then^Histrimmed,i.e.,^HT=^H.Otherwise,thereisavisuchthatvi=v0jforsomej6=i.Inthiscase,removetheedgefvi;v0igfromE(^H)andtheresulting^HTistrimmed. ThesetofgraphsHk=f^HTj^HTisatrimmedextensionofK1;kgiscalledthetrimmedextensionsetofK1;k. Notethatinthedenitionofextension,theverticesviarealldistinct,buttheverticesv0iarenotnecessarilydistinct,i.e.,itmaybethecasethatv0s=v0tforsomes6=t.Also,itispossiblethatsomev0i=vjforsomei6=j.However,v0i6=viforeveryi=1;:::;k. ItshouldbenotedthatthenumberofverticesofanextensionofK1;kisatmost2k+1,soHkisaniteset.Also,noticethatif^HT2Hk(followingtheabovelabelling)deg(vi)>1foreveryi=1;:::;kandallverticesof^HTareofdegreeatmostk. Thereasondeg(vi)>1foreveryi=1;:::;kisthateithertheedgesfv;vigandfvi;v0igareinE(^HT),orfv;vigandfvi;vjgareinE(^HT)forsomej6=i.Toseethatallverticesin^HTareofdegreeatmostkconsiderthefollowing.Supposethereisavertexuwithdeg(u)>k.Then,v6=u.Also,vi6=uforeveryi=1;:::;k,sinceallvi'saredistinctandtherecanbeatmostk1edgesincidentwithviandv0jforallj6=i(inthiscasetheedgefvi;v0igisremovedsotheyarek1)plustheedgefv;vigthedegreeofvibecomesatmostk.Then,u=u0jforsomej.Ifu0j=uiforsomeithecaseisexplainedabove.Then,atmostu0j=u0sforallj;s=1;:::;kinwhichcasedeg(u)kagain.Hence,deg(u)kforeveryu2V(^HT).72

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(k-regulargraphs.)Letk3.LetF=ffK1;k+1gg.ConsidertheenforcingsetE=fE1;:::;Ek+2gwhereE1=(;;fP2g),E2=(;fP2g),E3=(P2;fK1;kg),E4=(P3;fK1;kg),Ei+2=(K1;i;K1;k)fori=3;:::;k1,andEk+2=(K1;k;Hk)whereHkisthetrimmedextensionsetofK1;k.Then,L(F;E)=fGjGisk-regularg. Proof.LetGbeak-regulargraph.Obviously,Gcon Ek+2.Otherwise,thereexistiandj,suchthatvi=v0j.Considerallsuchviandremovefrom^Halledgesfvi;v0igforthesevi's.Then,^Hbecomestrimmed,i.e.,^H2Hk.Hence,^K1;kisembeddedin^HTforsome^HT2Hk.Therefore,Gsat Ek+2. Conversely,letG2L(F;E).SinceGcon E3theedgeehasatleastonevertexofdegreek.Ifdeg(u)=k,thenGis3-regular.Ifnot,considerthecopy^K1;kofK1;kinGinwhiche=fv;v1gwherevisthecentralvertexof^K1;kandu=v1(resp.u0=v).SinceGsat Ek+2,itfollowsthatdeg(v1)>1.Letdeg(v1)=s,where1
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E1;E2andsinceitis3-regular,Gsat E3;E4.Let^K1;3beacopyofK1;3inG.Letvbeitscentralvertexandv1;v2;v3bethethreedistinctverticesadjacenttov.SinceGis3-regular,thereexistedgesfv1;v01g,fv2;v02g,andfv3;v03g,withv016=v,v026=v,andv036=v.Then,eitherthereisaiforwhichthereisaj6=isuchthatvi=v0jornot.Ifthereissuchi,then^K1;3isenclosedineitherH1orH2inG(seeFigure5.9).Otherwise,noneofthevi'sequalsav0j,hence^K1;3isenclosedineitherH3,orH4,orH5inG.Hence,Gsat E5andthusG2L(F;E). Conversely,assumethatG2L(F;E).SinceGcon E3theedgeehasatleastonevertexofdegree3.E5requiresthattheothervertexofeiseitherofdegree2orofdegree3.Ifitisofdegree2,thenE3requiresthatitisofdegree3.Thus,Gis3-regular. 74

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FimpliesGcon F0whereF0=fD4g.Conversely,ifD4Gthenobviously,Gcon F.So,Gcon FifandonlyifGcon F0.Therefore,FcanbereplacedwithF0,whichinessencereducesFtoF0.75

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InrelationtotheGUgraphfromthepreviouschapter,aforbidderissubgraphfreeifforeverytwographsH;Kfromthisforbidder,thereisnopathfromHtoKorviceversaintheGUgraph. Whenreducingaforbidderbydiscardinggraphswhicharesubgraphsofothergraphsinthatforbidder,thenewlyobtainedforbidderismaximalinsomesense.Everygraphinamaximalforbidderdoesnothavesubgraphsinthatforbidder.Aformaldenitionfollows.Denition6.1.3 ItisclearthatFmaxissubgraphfree.Inaddition,foreveryFthereisanalgorithmforndingFmax.GivenaforbidderF,theforbidderFmaxisunique. UsingtheGUgraph,FmaxcanbefoundbyconsideringallverticesfromFandremovingallverticesKfromwhichthereisapathfromKtosomevertexHinF.Lemma6.1.4 Proof.LetF;F;HandKsatisfytheconditionsofthelemma.LetF0=(FnfFg)[fF0g.ItisobviousthatL(F0)L(F).LetGcon F0.IfL=H,thensinceH6G,itfollowsthatK6G.Hence,Gcon F0andL(F)L(F0). Subgraphfreeisanormalform,asstatedinthenextlemma.76

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Proof.LetFbegiven.ConsiderFmax=fFmaxjF2Fg.Clearly,forallF2FandforallgraphsGwehavethatFsub Ingeneral,sub Proof.AssumethatF1andF2aregivenandaresubgraphfree.Letsub Theabovelemmacan,also,beobservedthroughtheGUgraph.Theonlyverticesthatwillnotchangesub Somesubgraphfreeforbidderscanbereducedevenfurtherasshowninthefollowingexample.77

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TheaboveforbidderFcouldbereducedbecauseeveryconnectinggraphofF0containsK1;3asasubgraph.Consideranotherexample.Example6.1.8 InthegraphGU,thiscanbeobservedbynoticingthatforeverySsuchthatthereisapathfromK4toSandapathfromC5toS,thereexistsapathfromK1;4toS. Thenextlemmageneralizesthisreduction.Lemma6.1.9 Proof.LetF0=(FnfFg)[fF0g.ItisobviousthatL(F0)L(F).LetGcon F0.Otherwise,H6G,whichimpliesthatG62C(F0).Hence,Gcon F0.Therefore,L(F)L(F0)andFF0. Theabovelemma,also,followsfromProposition5.2.8.LetGcon F.IfGncon F0thenG2C(F0)andthereisaS2Cmin(F0)suchthatSG,butfromProposition5.2.8itfollowsthatS2Cmin(F)whichimpliesthatFsub F.Therefore,Gcon F0. ThereductionsfromLemma6.1.9canbeobservedthroughtheGUgraph.Con-siderthegraphsfromFasverticesinGU.IfforsomeH2FitholdsthatforallSforwhichthereisapathfromallK2FnfHgtoS,thereis,also,apathfromHtoS,thenHcanberemovedfromFwithoutchangingtheforbiddingfamily. AforbidderforwhichreductionsofthetypeinLemma6.1.9arenolongerpossibleiscalledconnectingfree.78

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IntermsofthegraphGU,aconnectinggraphSofaforbidderFisavertexSinV(GU),suchthatthereisapathfromeachH2FtoS.AforbidderFisconnectingfree,ifforeveryH2FthereisavertexS,suchthatforeveryK2FnfSg,thereisapathfromKtoS,butthereisnopathfromHtoS.Proposition6.1.11 Proof.LetFbeaconnectingfreeforbiddingset.LetF2FandH;K2F.SinceFisconnectingfree,itfollowsthatthereisaconnectinggraphS2C(FnfHg)suchthatHS,whichimpliesthatHK.Similarly,KS.Hence,Fissubgraphfree.Example6.1.7showsthattheconversedoesnothold. ThealternateproofbelowemploystheGUgraph. Theaboveproposition,also,followsfromProposition5.2.12.IfFisconnectingfree,theneveryF2FisconnectingfreeandbyProposition5.2.12Fissubgraphfree.Hence,Fissubgraphfree.Proposition6.1.12

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F0itfollowsthatGncon F0,i.e.,G2C(F0).Similarly,C(F0)C(F).Hence,C(F)=C(F0). Proof.FollowsfromthefactthatGnconFifandonlyifGncon F0. Thefollowinglemmashowsthatgivenaforbidderthereexistsanequivalentforbidderthatisconnectingfree.Lemma6.1.14 Proof.LetFbeaforbidder.IfFisconnectingfree,thelemmafollows.Otherwise,thereisaH2FsuchthatHSforeveryS2C(F).ConsiderF1=FnfHg.IfF1isconnectingfree,letF1=Ffree.Otherwise,thereisaH12F1suchthatH1SforeveryS2C(F1).ContinuethiswayandconsiderthesequenceF=F0;F1;:::.SinceFisnite,eventuallyanFkisreachedthatisconnectingfree.NotethatF0!F1!:::!FkandFi+1=FinfHig.ByProposition5.2.8Cmin(Fi)=Cmin(Fi+1)foreveryisoCmin(F)=Cmin(Ffree).Then,byProposition6.1.12andProposition6.1.13itfollowsthatFFfree. ThecorollarybelowfollowsfromLemma6.1.14.Corollary6.1.15

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Proof.LetFbegiven.FromLemma6.1.14,foreveryF2FthereisFfreesuchthatFfreeisconnectingfreeandGcon FifandonlyifGcon Ffree.DeneF0=fFfreejF2Fg.ItisclearthatF0isconnectingfreeandthatFF0. Anotherwaytoreduceredundancyistoremoveentireforbiddersthataresu-peruous.Considerthefollowingexample.Example6.1.17 ThekeyintheprecedingexampleisthatthesetofsubgraphsofeveryforbiddercontainsC3andmorepreciselythesetofsub Thenextlemmaisageneralizationofexample6.1.17.Lemma6.1.18 Proof.ItisclearthatL(F)L(FnfF2g).SupposeGcon Thefollowinglemmaisageneralizationoflemma6.1.18whichallowstheremovalofa(possiblyinnite)setofforbidders,ratherthanjustoneforbidder.Lemma6.1.19

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F.Otherwise,thereisaF02F0suchthatF06=Fandsub FandL(F0)L(F). Proof.LetFbeaforbiddingset.Becauseoflemma6.1.5assumethatFissubgraphfree.DeneF0=fF2Fjthereisno^F2Fsuchthat^F6=Fandsub NoticethatF0Fbydenition.TherestoftheproofshowsthatforeveryF2FthereexistsaF02F0withsub 82

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Theprecedingexampleshowsthatthenotionofsubgraphincomparableneedstobegeneralizedtoincludenotonlythesetofsubgraphsofaforbidder,butalso,thesubgraphsofconnectinggraphsofthatforbidder.Lemma6.1.23 Proof.LetF;F1,andF2beasintheconditionsofthelemma.ItisobviousthatL(F)L(FnfF2g).LetG2L(FnfF2g).SinceGcon F1,itfollowsthatG62C(F2),i.e.,Gcon F2. Theabovelemmaisgeneralizedbelowtoallowremovalofpossiblyinnitelymanyforbidders.Lemma6.1.24 Proof.Obviously,L(F)L(F0).LetG2L(F0)andF2F.SinceG62C(F0)itfollowsthatG62C(F).Hence,Gcon Fandthelemmafollows. RecallthatforagraphGandaforbidderF,eitherGcon ForG2C(F).TwoforbiddersF1andF2areequivalentifandonlyifGcon F1impliesGcon F2andviceversa.Hence,thefollowingremark.83

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So,ifF1andF2arenotequivalent,eitherthereexistsaconnectinggraphS2C(F1)suchthatF26sub IntermsoftheGUgraph,F1andF2areconnectingincomparableifthereisavertexSinGUwithapathfromeveryH2F1toS,suchthatthereisaK2F2fromwhichthereisnopathtoSandviceversa.Proposition6.1.27 Proof.LetFbeconnectingincomparable.LetF1;F22FwithF16=F2.SinceFisconnectingincomparable,thereexistsT2C(F2),suchthatT62C(F1).ThisimpliesthatF16sub

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NoticethatF0Fbydenition.TherestoftheproofshowsthatforeveryF2FthereexistsaF02F0withC(F)C(F0).LetF2F.IfF2F0,theclaimistrue.Ifnot,ithasbeenexcludedfromF0bydenition.Eitherthereissome~F2F0suchthatF~F,inwhichcasetheclaimistrueorFhasbeenexcludedfrom(^F)inwhichcasethereisaF1suchthatC(F)C(F1)andF6F1.IfF12F0theclaimistrue.Ifnot,eitherF1F2forsomeF22F0inwhichcasetheclaimistrueorthereisF26F1withC(F1)C(F2).ContinuethiswayandconsiderthesequenceofforbiddersF0;F1;F2;:::whereF0=F,Fi+16Fi,andforeachi0itholdsthatC(Fi)C(Fi+1).SinceFi6Fi+1fromRemark6.1.25itfollowsthatC(Fi)6=C(Fi+1),therefore,C(Fi)C(Fi+1).LetS2C(F).IfthesequenceF0;F1;F2;:::isinnite,itfollowsthatSisaconnectinggraphofinnitelymanyforbidders,whichcontradictsthefactthatSisnite.Therefore,aFk2F0isreachedsuchthatC(F)C(Fk).Thus,theconditionsofLemma6.1.24aresatisedanditfollowsthatFminF. BecauseTheorem6.1.21showsthatconnectingfreeandconnectingincomparableisanormalformconsiderthedenitionbelow.Denition6.1.30

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Theconverse,however,isnotnecessarilytrueevenifFisinreducednormalform.ConsiderF=ffC3;C4ggandD42C(fC3;C4g.LetF0=ffD4gg.Then,thegraphG=SC3P4C4fromExample5.2.3andFigure5.1issuchthatGcon Proof.LetGcon Proposition6.1.31nowfollowsfromtheaboveproposition.Proposition6.1.32provesthateveryforbiddingsetisequivalenttoaforbiddingsetconsistingofsin-gletonforbiddersonly.Hence,thefollowingtheorem.Theorem6.1.33

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Proof.LetF=fKg2F.ItisclearthatL(F)L(FnfFg).Obviously,K62L(F).IfH6=F,thenH6sub Proof.LetFbegivenandconstructF0asinTheorem6.1.33.Then,fromF0constructaconnecting(subgraph)incomparableforbiddingsetF00asinTheorem6.1.29.Then,byProposition6.1.35,F00isminimal,i.e.,L(F00)L(F00nfFg)foranyF2F00. Proof.LetF2F1.SinceF1F2,itfollowsthatthereisaH2F2suchthatHsub Proof.LetFbegivenandletF0bethestrictforbiddingsetconstructedasinTheorem6.1.33consistingofsingletonforbiddersofallminimalconnectinggraphs87

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NotethatifanenforcingsetEcontainsabruteenforcer(;;Y),thenifG2L(E),itfollowsthatGhasatleastonevertex,henceGsat Thenextpropositiongeneralizesthistypeofredundancy.Proposition6.2.4 Proof.Obviously,L(E)L(E0).LetGbeinL(E0).IfX6GthenGsat Theaboveresultisextendedtoremovingasubsetofredundantenforcersofthistypeinsteadofjustonesuchenforcer.88

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Proof.Obviously,L(E)L(E0).LetG2L(E0)and(X;Y00)2E00.IfX6G,thenGsat AgraphinYcanbeasubgraphofanothergraphinY,whichinsomecasesleadstothetypeofredundancyexaminednext.Proposition6.2.6 Proof.LetGsat Thefollowingexampleshowsthateveninaveryrestrictivecase,itisstillverydiculttodeneredundancyforenforcerswithdierentrstcomponents.Example6.2.8

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Ageneratedgraphgm(X)iscalledminimal,ifnopropersubgraphofitisageneratedgraph. LetEbeanenforcingsetandletX2E(1).DenotethefamilyofgeneratedgraphsofXwithrespecttoEwithGEXorsimplyGXwhenEisunderstood.ThefamilyofminimalgeneratedgraphsofXwithrespecttoEisdenotedbyMEXorsimplyMXwhenEisunderstood.LetM(E)=[X2E(1)MX. Thenextpropositionfollowsdirectlyfromthedenitionofgeneratedgraphs.Proposition6.2.10

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ConsiderE=f(P3;fC3g)g.TheminimalgeneratedgraphofP3is(isomorphicto)C3.ItisniteanditsatisesE.K4isageneratedgraph,butnotminimalsinceC3K4.(b)LetE0=f(P3;fP4g);(P4;fP5g);:::;(Pn;fPn+1g);:::g.Theminimalgener-atedgraphgm(P3)isinniteandnographinL(E)containsP3asasubgraph.Thus,L(E)=f;;;P2g.Example6.2.12 Theaboveexampleshowsthataninniteenforcingsetisequivalenttoaniteenforcingset.Thisraisesthequestionwhetherthereareinniteenforcingsetsthatarenotequivalenttoanyniteenforcingsetandthefollowingtwoexamplesandpropositionprovideanarmativeanswer.Example6.2.13 Theabovetwoexamplesprovethefollowingproposition.91

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Thisfactshowsthatthereisanon-nitaryenforcingsetthatisequivalenttoaniteenforcingset. Fromtheaboveexamplesitfollowsthat^EE0,where^EisasinExample6.2.17andE0isasinExample6.2.12.Denition6.2.18 Thefollowingpropositionshowsthatredundantenforcerscanbeerasedfromtheenforcingset.Proposition6.2.20 Proof.ItisclearthatL(E)L(E0).LetG2L(E0).IfX06G,thenG2L(E).AssumeX0G.Then,XGwhichimpliesthatG=g(X)forsomeg(X)2GE0X.92

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ThefollowinglemmashowsthataninnitenitaryenforcingsetwithniteM(E)musthaveaninnitegeneratedgraph.Lemma6.2.21LetEbeinniteandnitary,suchthatM(E)isnite.Thenthereexistsaninnitegeneratedset. Proof.SinceM(E)isnite,thereisanitenumberoffamiliesofminimalgeneratedgraphsMX.DenotethesefamiliesbyM1;M2;:::;Mk,i.e.,M(E)=Ski=1Mi.SincethereareinnitelymanydistinctX's(duetoEbeinginnite)andnitelymanyMi's,theremustexistatleastoneMjsuchthatforinnitelymanyX'sinE(1),itholdsthatMX=Mj.Letgm(X)2Mj.Sincegm(X)isageneratedgraphforinnitelymanyX's,itfollowsthatgm(X)containsalltheseX'sassubgraphs.Hence,gm(X)isinnite.(InfactallgeneratedgraphsinMjareinnite.) 93

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InthischapterapartiallyorderedsetisusuallydenotedbyPandthepowersetofPwithP(P).TheorderinPisdenotedby.AsetofelementsfromPiscalledasubposet.Thus,subposetsareelementsinP(P).Apartiallyorderedsetdoesnotnecessarilycontainasmallestelement,butincaseitdoes,thesmallestelementisdenotedby.Inthiscase,pforeveryp2P.AsubposetLPisachainifforanyp;q2Litholdsthateitherpqorqp.AsubposetKPisanantichainifforanyp;q2Kwithp6=qneitherp
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Someexamplesofposetsinclude:(i) ThesubsetsP(S)ofasetSorderedbyinclusionanddenotedby(P(S);).(ii) ThewordsAoveragivenalphabetAwithsubwordorderdenotedwith(A;sub Graphswithsubgraphorder(G;)whereGisthesetofsimpleconnectedgraphsandHGifandonlyifHisasubgraphofG.(iv) Naturalnumberswithdivisibility(N;j)wherepqifandonlyifpdividesq(pjq). Inalloftheaboveexamples2PandallelementsofParenite.Theposetofintegers(Z;)doesnothaveasmallestelementandcontainsinniteelements.7.1fe-FamiliesasSetsofSubposets

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FifF6sub FforallF2F. ForaforbiddingsetFthefamilyofF-consistentsubposetsisL(F)=fLjLcon ThefamilyL(F)issaidtobedenedbytheforbiddingsetF.AfamilyofsubposetsLiscalledanf-family,ifthereisaforbiddingsetFsuchthatL=L(F).ThemaximalsubposetsaredenedwithrespecttoinclusionandM(F)denotesthesetofmaximalsubposetsinL(F). ThecorrespondingboundaryobservationsfromChapter2holdforposets,aswell.Remark7.1.2(i) Inadditionto(A;sub Again,Fiscalledastrictforbiddingsetifitcontainssingletonforbiddersonly.Example7.1.5

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AsubposetLissaidtosatisfyanenforcer(X;Y)ifXLimpliesY\L6=;.ForanenforcingsetEthefamilyofallsubposetsLthatsatisfyEisdenotedbyL(E).AfamilyofsubposetsLiscalledane-familyifthereexistsanenforcingsetEsuchthatL=L(E). InthecasethatX6L,Lissaidtosatisfytheenforcertrivially.EnforcersinwhichX=;arecalledbrute.Inthiscase,anelementfromYhastobeinLinorderforLtosatisfytheenforcer.Anenforcer(X;Y)forwhichX\Y6=;iscalledtrivial,sinceitisalwayssatised.Thenotationanddenitionforgeneratedsubposetsisextendedaccordingly.Remark7.1.7(i) Inwhatfollows,unlessotherwisestated,allenforcersarenon-trivial.Denition7.1.8 Insomesense,strictenforcers\force"YintothesubposetL.Considerthefollowingexampleofastrictenforcingset.Example7.1.9 Respectively,thetwonotionsofforbiddingandenforcingarecombinedintothefollowingdenition.97

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AfamilyofsubposetsLiscalledaforbidding-enforcingfamilyorfe-family,ifthereexistsafe-system(F;E),suchthatL=L(F;E).Example7.1.11 Twosetsofforbidders(ortwoenforcingsets,ortwoforbidding-enforcingsys-tems)areequivalentiftheydenethesamefamilyofsubposets.Theequivalencerelationisdenotedby. FromtheabovedenitionsitfollowsthatthereisnoenforcingsetEsuchthatL(E)=;.NeitherthereisaforbiddingsetFsuchthatL(F)=;.Remark7.1.12 Theimmediatepropertiesoffe-systemsonlanguagescanbeextendedtofamiliesofsubposetsandcaneasilybeveried.AposetPisassumed.Proposition7.1.14

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Thenormalformsforfe-systemsonformallanguagesfrom[8,9,40]followdirectlyforsubposetswithcertainrestrictions.Example7.1.15 Theaboveexampleshowsthattheforbidderf2;3;6gisredundant.Itisnotsubelementfreebecause22sub Thisdenitiongeneralizesthesubwordfreeconditionforforbiddingsetsonlan-guagesandLemma11.2in[40]holds,i.e.,foreveryforbiddingsetthereisanequivalentsubelementfreeforbiddingset.99

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Notonlyelementswithinaforbiddermayberedundant,butalsoforbiddersthemselvescanberedundant.Thefollowingexampleillustratessuchredundancy.Example7.1.17 Thereasonfortheaboveredundancyisthatsub Thefollowingdenitiongeneralizesthenotionofsubwordincomparableforbid-dersfrom[40].Denition7.1.19 Someresultsforfe-systemsonlanguagesin[40]wereprovedusingthefactthatsub

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Again,aforbiddingsetisinminimalnormalform,ifitisbothsubelementfreeandsubelementincomparable.Intheeventthatsub From[40],itholdsthatFmax Somepropertiesoflanguageenforcingsetsholdforposets,aswell,buttheirproofsneedtobeadjusted.Considerthefollowingtwoobservationsfrom[40].Lemma7.1.22 Proof.LetE=f(X1;Y1);:::;(Xn;Yn)g.LetX=[ni=1XiandY=[ni=1Yi.Then,L=X[YissuchthatLsat Proof.ConsidertheenforcingsetE=f(;;fp1g);(fp1g;fp2g);:::g.TheneveryL2L(E)containstheinnitechainp1;p2;:::;pn;:::,i.e.,isinniteandbyLemma7.1.22aniteE0suchthatE0Edoesnotexist. Someobservationsforredundancyofenforcingsetsforlanguagescanbeextendedtoposetsdirectly(see[40]).Thus,trivialenforcers(forwhichX\Y6=;)provideanobviousredundancy.Also,iftheenforcingsetcontainstwoenforcers(X;Y)and(X0;Y0)suchthatXX0andYY0,then(X0;Y0)isredundant.101

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TheprocedureofevolvingthroughE-extensionswasproposedforenforcingoflanguagesin[8,9,40].Itisgeneralizedtoposets.Denition7.1.24 LetEbeanenforcingset.DeneE(1)=fXj(X;Y)2Eg.Thedenitionofgeneratedsetsforlanguagescan,also,beextendedtoposets.Denition7.1.25 Ageneratedsetgm(X)iscalledminimal,ifnopropersubsetofitisageneratedset. ThenotationfromChapter2isusedhere,aswell.LetEbeanenforcingsetandletX2E(1).DenotethefamilyofgeneratedsetsofXwithrespecttoEwithGEXorsimplyGXwhenEisunderstood.ThefamilyofminimalgeneratedsetsofXwithrespecttoEisdenotedbyMEXorsimplyMXwhenEisunderstood.LetM(E)=[X2E(1)MX.Remark7.1.26 AllresultsfromSection2.5follow. Thefollowingdenitiongeneralizesthenotionofredundancy.102

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Inparticular,ifXX0andYY0,then(X0;Y0)isredundant.Redundantenforcerscanberemovedfromtheenforcingsetwithoutchangingthee-family(seeSection2.5). Asinthecaseoflanguages,M(E)canbeniteorinnitewithniteorinnitegeneratedsets.IfEisnite,thenconstructL=X[YasintheproofofLemma7.1.22andobservethateverygm2M(E)issuchthatgmL.SinceLisnite,M(E)consistsofnitelymanyniteminimalgeneratedsets. InthecasethatEisinniteandnotequivalenttoanynitesetandM(E)isnite,thenLemma2.5.13holdsandthereisaninniteminimalgeneratedsetinM(E).7.2fe-SystemsDeningaSingleSubposet F,ifF6sub FforallF2F.Thisisdenotedbywcon ForaforbiddingsetFthefamilyofF-consistentelementsisL(F)=fwjwcon ThefamilyL(F)issaidtobedenedbytheforbiddingsetF.AsubposetLiscalledanf-family,ifthereisaforbiddingsetFsuchthatL=L(F).103

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Weakenforcing.AnenforcingsetEisapossiblyinnitefamilyoforderedpairs(X;Y),whereXandYarenitesubposetsofP,suchthatY6=;.Theelements(X;Y)ofanenforcingsetEarecalledenforcers. Anelementwissaidtoweaklysatisfyanenforcer(X;Y)ifXsub InthecasethatX6sub

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Strongenforcing.AnenforcingsetEisapossiblyinnitefamilyoforderedpairs(X;Y),whereXisanelementinPandYisanitesubposetofP,suchthatY6=;andX
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Thefollowingremarkholdsforbothweakandstrongenforcing.Remark7.2.11(i) If2P,then2L(E)foreveryEthatdoesnothavebruteenforcers.Denition7.2.12 Insomesense,strictenforcers\force"YintothesubelementsofwwheneverXsub Thefe-systemsdenitioncanbeextendedtoposetsaccordingly.Denition7.2.14 AsubposetLiscalledaforbidding-enforcingfamilyorfe-family,ifthereexistsafe-system(F;E),suchthatL=L(F;E). Twosetsofforbidders(ortwoenforcingsets,ortwoforbidding-enforcingsys-tems)areequivalentiftheydenethesamefamilyofelements(thesamesubposet).Theequivalencerelationisdenotedby.TwoforbiddersFandF0(similarlytwoenforcersEandE0)areequivalentdenotedbyFF0(EE0),ifallelementsthatareconsistentwithF(satisfyE)arealsoconsistentwithF0(alsosatisfyE0)andviceversa.106

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Thefollowingremarkholdsforbothweakandstrongenforcing.Remark7.2.16 Thenexttwoexamplesillustratesuchposetsandholdforbothweakandstrongenforcing.Example7.2.17

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However,ifunionisdenedinasimilarway,i.e.,sub

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Inthissection,thediscussionofconnectinggraphsfromChapter5isextendedtoposets.Thenotionof\connectinggraphs"isgeneralizedto\upperbounds".Denition7.3.1 So,foraforbidderFandanelementw2P,eitherwcon Forw\connects"(isanupperboundof)F.Foreveryw2C(F)thereisas2Cmin(F)suchthatsw. Itisobviousthatifanelementisanupperboundofanitesetofelements,itisalsoanupperboundofeverysubsetofthissetofelements.ThisfactisadirectextensionofProposition5.2.6.Proposition7.3.2 Inthecaseofnaturalnumberswithdivisibility,foreverynitesetofnumbers,theminimalconnectingelementisuniqueanditisequaltotheleastcommonmultiple(LCM)ofthenumbersintheset.Remark7.3.4

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Consideraspecialcase,whichisadirectgeneralizationofProposition5.2.8.Proposition7.3.6 TheabovedenitionsaysthatifX=fpg,thenpcannotbeanextensionofitself,eventhoughitisanupperboundofX,butq2Psuchthatq6=pandpqisanextensionofp.So,everyextensionisanupperbound,buttheconversedoesnothold.Denition7.3.8 Proof.LetPbeextendable.ByDenitions7.3.7and7.3.8,ifXisanitesubposetofP,thenthereisanextensionX1ofXthatisanupperboundofX.Hence,Pisweaklyextendable.ConsideranitesetSandletP=(P(S);).Then,SisanupperboundforeverynitesubsetXP(S),butSdoesnothaveanextension. Theposet(A6n;sub TheforbiddingthroughenforcingobservationsfromChapter5canbegeneralizedforextendablePconsistingofniteelementsonly.110

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Theaboveexampleshowsthattheforbidderfp1;p2;qgisredundant.Itisnot\subelementfree",becausep1qandp2q.Denition7.4.2

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Fifandonlyifnisnotdivisibleby6. \Subelementfree"isanormalform.TheproofofthefollowinglemmaisadirectgeneralizationoftheproofofLemma6.1.5.Lemma7.4.4 Thefollowingexampleshowsthataforbidderthatisanantichainisnotneces-sarilyconnectingfree.Example7.4.7 F0,thenwcon F.Conversely,ifwcon Feither6-w,or10-w,or15-w.Inthersttwocaseswcon F0.Suppose15-w.Then,either3-wor5-w.Intherstcase6-wandinthesecond10-w.Therefore,wcon F0.112

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Notonlyelementswithinaforbiddercanberedundant,butalsoforbiddersthemselvescanberedundant.Next,thesubelementincomparablenormalform,whichisageneralizationofthenormalformsforgraphf-systemsfromChapter6andlanguagef-systemsfrom[40]isdiscussed.Example7.4.14

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Thefollowinglemmastatesthatsubelementincomparableisanormalform.Lemma7.4.18

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Ingeneral,iftwoforbiddersFandHshareacommonminimalupperboundsitdoesnotnecessarilyfollowthatFH.Also,evenforaminimalupperboundsofFitdoesnotnecessarilyfollowthatFfsg.Forexample,considerfaa;bbgover(A;sub However,for(N;j)itholdsthatforeveryF,Ffsgfortheminimalupperbound(LCM)sofF.Proposition7.4.28 Proof.Letwcon F. Becauseoftheaboveproposition,allforbiddersover(N;j)canbereplacedwithsingletons,i.e.,onecanobtainstrictforbiddingsetsonly.Becauseofsubelementincomparablenormalform,iffsgandfksgaretwoforbiddersinF0,thenfksgisredundant.So,fksgcanberemovedfromF0.Furthermore,sinceallforbiddersaresingletons,parenthesescanbeomitted,i.e.,F0=fh1;:::;hn;:::g,whereforeveryiandeveryjitholdsthathi-hj.So,anumberw2L(F)ifhi-wforeveryi1. Astrictforbiddingsetisalwaysconnecting(subelement)freeandsubelementincomparablecoincideswithconnectingincomparable.So,theforbiddingsetde-scribedabovefor(N;j)isindeedminimalandunique.116

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Proof.Letwcon ToseethatequalitydoesnotnecessarilyholdconsideronceagainF=faa;bbgover(A;sub Proof.Letwcon Proposition7.4.30provesthateveryforbiddingsetisequivalenttoastrictfor-biddingset.Thisfactisstatedformallybelow.Theorem7.4.31

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However,forposetsinwhicheverynitesubposethasauniqueminimalupperboundauniqueminimalnormalformcanbeachieved.Theposet(N;j)isonesuchexample.Proposition7.4.32 Proof.Let^F=ffsFgjF2Fg.LetF0=ffsg2^Fjthereisnofhg2^Fwithhsg.Obviously,F0isconnectingfreeandconnectingincomparable.FromProposition7.4.30itfollowsthatFF0.ConsiderF00=F0nfsg.Then,L(F0)L(F00).Clearly,sncon LetF1andF2aretwosuchminimalforbiddingsetswithF1F2andletF2F1.Then,F=fsgforsomesands62L(F2).IffollowsthatthereisF0=fhg2F2suchthatF0sub 118

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Thefollowingdiscussionisforweakenforcingsetsonly.Normalformsforen-forcingingraphfe-systemscanbegeneralizeddirectlyforstrongenforcingsets.Remark7.5.1 Lemma11.14from[40]holds.Lemma7.5.2 Ageneratedelementgm(X)iscalledminimal,ifforeverygeneratedelementp(X)pgm(X),impliesthatp=gm(X). LetEbeanenforcingsetandletX2E(1).DenotethesubposetofgeneratedelementsofXwithrespecttoEwithGEXorsimplyGXwhenEisunderstood.The119

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Theabovedenitionofgeneratedelementsallowstheelementsgm(X)tobesuchthatsub RedundancyforenforcingsetscanbedenedasinChapter2.Denition7.5.6 ThefollowinglemmafromChapter2holdsinthiscase,aswell.Itshowsthatredundantenforcerscanbeerasedfromtheenforcingset.120

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Asinthecasewithfamilyofsubposets(andfamiliesoflanguages),M(E)canbeniteorinnitewith\nite"or\innite"elements.Example7.5.9 ThefollowinglemmashowsthataninnitenitaryenforcingsetwithniteM(E)musthaveaninnitegeneratedelement.Proposition7.5.10 Proof.SinceM(E)isnite,thereisanitenumberoffamiliesofminimalgener-atedelementsMX.Denotethese(families)setsbyM1;M2;:::;Mk,i.e.,M(E)=Ski=1Mi.SincethereareinnitelymanydistinctX's(duetoEbeinginnite)andnitelymanyMi's,theremustexistsatleastoneMjsuchthatforinnitelymanyX'sinE(1),wehaveMX=Mj.Letgm(X)2Mj.Sincegm(X)isageneratedelementforinnitelymanyX's,itfollowsthatgm(X)containsalltheseX'sinsub 121

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Proof.SupposethatM(E)isniteandproceedasintheaboveproof.Thisimpliesthataninnitegeneratedelementexists,whichcontradictsthefactthatPcontainsniteelementsonly.Hence,M(E)isinnite. 122

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Thefe-systemsdenitionspresentedinthepreviouschapterswereusedtode-ne,ratherthanderivethefamilyofstructuresthatobeysthefe-system.Incom-puting,thecomputationprocessbeginswithasetofinitialconditions.Therefore,additionaldenitionsareintroducedtoobtainafamilyofstructuresderivedbyanfe-system. Anfe-systemgeneratedbyobjectKisapair=(K;)whereKisanobject(i.e.,language,graph,subposet,etc.)andisanfe-system.OnecanconsiderKasasetorstructuredeningtheinitialconditions.TheclassofobjectsdenedbyisL()=fLjKLandL2L()g.SupposeKandLaretwoobjectswhichgenerateKandLrespectively.Then,KLimpliesL(L)L(K).123

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Bondingrelationships(bothcovalentandweakhydrogenorionicbonds)inmoleculescanbedescribedthroughfe-systemsoflanguages.Asanillustration,thissectionconsidersthestructureofthedoublestrandedDNAandtheactionsofendonucleasesandligases. ADNAstrandcanbeconvenientlyrepresentedasastringoverasuitableal-phabet.ThebasicalphabetisA=fa;c;t;ggwherethesymbolsrepresentthefourdierentkindsofbasesofnucleotidesintheDNAmolecule:\adenine",\cytosine",\thymine",and\guanine".Watson-Crickcomplementaritydictatesthatapairswithtandgwithc.Forasymbolx2A,itsWatson-Crickcomplementisdenotedwithx.ThisnotationisextendedtowordsoverAwithwbeingthecomplementofw,wRbeingthereverseofw,andwRthereversecomplementofw.Forexample,ifw=aatcgathenw=ttagctandwR=tcgatt.Thus,wRisthecomplementofwR.Eachnucleotideconsistsofasugar,aphosphategroupandanitrogenousbase.AstrandofDNAisobtainedwhenacovalentbondisestablishedbetweenthephosphategroupofonenucleotideandthehydroxylgroup(fromthesugar)ofanothernucleotide.Thisleavesaphosphateononeend,standardlydenotedas50andahydroxylgroupontheotherend,standardlydenotedas30.DrivenbyWatson-CrickcomplementaritytwostrandsofDNAcanformadoublestrandedmoleculebyforminghydrogenbondsbetweencomplementarynucleotides.Abasepairofnucleotidescanberepresentedwithapairofsymbolsxx,wherex2AandxistheWatson-Crickcomplementofx.Similarly,apieceofdoublestrandedmoleculeisrepresentedasaconcatenationofsuchsymbols,say50aatcga3030ttagct50.Itiscustomarytowritetheupperstrandinthe5030direction,(whichorientsthelowerstrandinthe3050direction)andthusomitthe50and30fromthenotation.124

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EndonucleasesrecognizeshortsequencesofDNAandperform\cuts",whichareeitherbluntorleave\sticky"overhangs.Figure8.1leftshowsexamplesoftherecognitionsitesoftheenzymesBfaIandMseI.Eventhough,bothenzymeshavedierentrecognitionsitestheyleavethesame(singlestranded)50overhangta.Twostringswithcomplementaryoverhangscan\stick"togetherandformacompletelynewstring.Thus,amoleculewith50overhangtaontopattachestoalowerstrandoverhangat,byformingahydrogenbondbetweentandaandaandtrespectively.Aligasegluesthenicksatthepositionswherethemoleculeshavejoinedestablishingacovalentbondbetweentandthenucleotidenexttoitonthetopstrand.Similarly,thelowerstrand50-endofaformsacovalentbondwiththe30-endoftheadjoiningnucleotide.Theabovedescribedoperationof\cutting"and\pasting"withenzymesisknowninliteratureassplicing.Figure8.1rightshowsthetwostepsofDNArecombination. Considerthealphabet=fxx,x,x,xx,xx,x;xjx2Ag.Hereisusedtoindicateanexistingnickortoidentifythe50end.Theindicatesamissingnucleotide,soindicatetwomissingnucleotides.Thisalphabetissimilartotheoneusedin[8].Toeasethenotationtheinnerparenthesesareignoredwhenconcatenatingsymbols. AlanguageofvalidsingleordoublestrandedDNAbelongstothefamilyL()125

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Theprocessofligating,orclosingthe\nicks"isdenedwithE0=f(X=fwxyxyzg;X[fwxyxyzg),(X0=fuxyxyvg;X0[fuxyxyvg)jx;y2Aandu;v;w;andzarewordsoverg. CuttingwithBfaIcanbemodeledwithEBfaI=f(X=fwctaggatczg;Y=X[fwcgatg)jforeachpairofwordswandzg.Then,recombinationthatoccurswhentwopieceswithoverhangtaonefromBfaIandanotheronefromMseIannealcanbemodeledintwosteps.First,annealing(self-assemblybythehydrogenbonds)canbeexpressedbyEta=f(X=fwcgat;taatzg;X[fwctaagattzg)jforeachpairofwordswandzg.Then,theligationisobtainedbytheenforcersE0. NowconsideramolecularmixwithaninitialsetofDNAstrandsK,aset126

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LetSbeasetandP(S)itspowerset.Considertheposet(P(S);),whereisthesubsetrelation.Thek-colorabilityproblemaskswhethergivenagraphandanitesetofkcolorsitispossibletoassignonecolortoeachvertexinsuchawaythatadjacentverticeshavedistinctcolors.Suchassignmentofcolorsiscalledak-coloring. LetG=(V;E)beagraphwithnvertices,i.e.,V=fv1;v2;:::;vngandCbeasetofkcolorsi.e.,C=fc1;c2;:::;ckg.Ak-coloringwillbeviewedasasubsetofVC. ConsidertheposetP=(P(VC);)andthefollowingfe-system.Everyvertexshouldbeassignedexactlyonecolorwhichcanbedonebyacombinationofbruteenforcingandforbidding.ThebruteenforcingE=f(;;f(v;c1);(v;c2);:::;(v;ck)g)jv2Vgensuresthateveryvertexisassignedatleastonecolor.TheforbiddingsetF=ff(v;c);(v;c0)gjv2Vandc;c02Cwithc6=c0gallowsonlytheseverticesthatareassignedatmostonecolor. Also,notwoadjacentverticesshouldbecoloredthesame.Thisisobtainedwith127

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Let=(F[F0;E)beanfe-system.ThenGisk-colorableifandonlyifL()6=;.Furthermore,anysetK2L()containsexactlyonek-coloringofG.Notethatuptopermutationsofthecolors,allsolutionstothek-colorabilityproblemcanbeobtainedbyxingonevertexandonecolor.So,setv2Vandc2CandletK=f(v;c)g.ThenGisk-colorableifandonlyifL(K;)6=;.Furthermore,anysetK2L(K;)containsexactlyonek-coloringofG.128

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Morphismsarenaturalmapstoconsiderbetweenlanguages.Althoughthisworkcharacterizesthemorphismsthatmapf-familiestoextendedf-familiesandprovidesresultsaboutmorphicimagesoff-familiesande-families,thequestionofwhatmorphismsmapfe-familiesintofe-familiesremainsopen. Normalformsprovideafoundationforstudyinganytypeoffe-systems.Inlanguagefe-systems,[8,9,40]presentnormalformsforforbiddingsetsandforen-forcingsets.Thispaperprovidesnewnormalformsforlanguagefe-systems.Evenifaforbiddingsetisinminimalnormalformandanenforcingsetisinnitarynor-mal,whencombinedinafe-system,thesystemasawholemaybereducedfurther.Inthisrespect,any\interaction"betweentheforbiddingsetandtheenforcingsetofanfe-systemshouldbeinvestigatedfurther. Theposetofgraphswiththesubgraphorderwherefe-systemsdeneasubposet(classofgraphs)providesanewwayofclassifyinggraphs.Thedenitionsandprop-ertiesoffe-graphsledtoanewgraph-theoreticalnotion-connectinggraphsofaset129

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Thispaperstatessomenormalformsforforbiddingsetsandenforcingsetsforgraphfe-systems.Connectingfreeandconnectingincomparableareshowntobeminimalinsomecases.Continuingtheinvestigationofminimalanduniquenormalformsforforbiddingsetsisthenaturalnextstep.Enforcingsetsareinherentlydiculttostudy.Newnormalformsforenforcingsetswouldbeofinterest.Gener-atingsetsmayprovideanaturalrststepinthisdirection.Itisworthmentioningthatmanyothergraphfe-systemsmodelscanbedenedandinvestigated.Ifintheforbiddingdenitionthewordsubgraphisreplacedwithinducedsubgraphthenageneralversionofthehistoricalconceptofforbiddingsetsisobtained.Acompre-hensivelistofpapersconsideringsuchtypeofforbiddingwheretheforbiddingsetsarestrictisavailableat[18].Non-strictforbiddingsetsforinducedsubgraphshavenotyetbeendiscussedingeneral.Sincetherearealotofconditionsforhamiltonic-itydescribedthroughstrict\induced"forbiddingsets,amoregeneraldenitionofallowingnotonlysingletonenforcerscanprovidedirectgeneralizations.Otherversionsofenforcingcan,also,bepursued.Itwillbeinterestingtoknowwhetherrelaxingtheembeddingconditionforenforcers(\weak"enforcing)leadstonewcharacterizations.Also,restrictingtherstcomponentofanenforcertoconnectedgraphsmayleadtonewgraphfe-systemsproperties.Anothervariantofenforcingsetsthatcanbedenedisamodelwheretherstcomponentofanenforcerisasetofconnectedgraphs,notjustonegraph. AgeneralizationoftheweakversionofenforcingtoposetsispresentedinChapter7.Itcan,also,bedenedonsomespecicposetslikeAwithsubwordorder.ConsideringatopologyonA(see[34])mayenhancetheunderstandingoffe-130

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Althoughthegeneraltypesoffe-systemsshouldbestudiedinthecontextofposetsandevencategories,investigatingeachvariantoffe-systemsinaspecicposet(forexamplelanguages,words,graphs,groups,andmatrices)mayprovidefornewpropertiesoffe-familiesduetotheinherentstructureoftheposet.Investigat-ingfe-systemsintheposetofgeometricstructuresmadeofsmallbuildingblocksmayprovideacompletelynewapproachintostudyingcrystalsandself-assemblyprocesses. Thecomputingpotentialoffe-systemisyettobeinvestigatedindetail.Boththeoreticalandpracticalresultsshouldbepursued.131

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Forbidding and enforcing of formal languages, graphs, and partially ordered sets
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ABSTRACT: Forbidding and enforcing systems (fe-systems) provide a new way of defining classes of structures based on boundary conditions. Forbidding and enforcing systems on formal languages were inspired by molecular reactions and DNA computing. Initially, they were used to define new classes of languages (fe-families) based on forbidden subwords and enforced words. This paper considers a metric on languages and proves that the metric space obtained is homeomorphic to the Cantor space. This work studies Chomsky classes of families as subspaces and shows they are neither closed nor open. The paper investigates the fe-families as subspaces and proves the necessary and sufficient conditions for the fe-families to be open. Consequently, this proves that fe-systems define classes of languages different than Chomsky hierarchy. This work shows a characterization of continuous functions through fe-systems and includes results about homomorphic images of fe-families.This paper introduces a new notion of connecting graphs and a new way to study classes of graphs. Forbidding-enforcing systems on graphs define classes of graphs based on forbidden subgraphs and enforced subgraphs. Using fe-systems, the paper characterizes known classes of graphs, such as paths, cycles, trees, complete graphs and k-regular graphs. Several normal forms for forbidding and enforced sets are stated and proved. This work introduces the notion of forbidding and enforcing to posets where fe-systems are used to define families of subsets of a given poset, which in some sense generalizes language fe-systems. Poset fe-systems are, also, used to define a single subset of elements satisfying the forbidding and enforcing constraints. The latter generalizes graph fe-systems to an extent, but defines new classes of structures based on weak enforcing. Some properties of poset fe-systems are investigated. A series of normal forms for forbidding and enforcing sets is presented.This work ends with examples illustrating the computational potential of fe-systems. The process of cutting DNA by an enzyme and ligating is modeled in the setting of language fe-systems. The potential for use of fe-systems in information processing is illustrated by defining the solutions to the k-colorability problem.
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