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Recognizable languages defined by two-dimensional shift spaces

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Recognizable languages defined by two-dimensional shift spaces
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Pirnot, Joni Burnette
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Recognizable picture languages
Dot systems
Two-dimensional finite automata
Transitivity
Doubly periodic
Dissertations, Academic -- Mathematics -- Doctoral -- USF
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ABSTRACT: There are numerous connections between the theory of formal languages and that of symbolic dynamics. In each, the transition from one dimension to two dimensionsis accompanied by much difficulty due in large part to the emptiness problem, which is related to the presence (or lack thereof) of periodic points and is known to be undecidable. Here, we focus on two-dimensional languages that have the property that all blocks allowed by the language can be extended to a configuration of the plane satisfying the structure of the language; for such languages the emptiness problem is not an issue. We first show that dot systems may be associated with two-dimensional languages having this property, so that we might employ these languages as varied examples. We next define a new type of finite automaton and with it, a tool for recognizing two-dimensional "strings" of data. It is then shown that these automata correctly represent the sofic shift spaces that result from the application of block maps to shifts of finite type. Thereafter, these automataare utilized to investigate properties of transitivity in the two-dimensional languages that they represent. More specifically, new definitions for different types of two-dimensional transitivity are adapted from topological dynamics and then illustrated through the use of dot systems. The appearance of periodic points in the languages represented by these automata is also explored, with a main result being that the existence of a periodic pointis guaranteed under certain conditions. Finally, issues of equivalence are introduced in the two-dimensional setting with regards to formal languages (syntactic monoids) and symbolic dynamics (the follower sets of a graph representing a sofic shift space).
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Dissertation (Ph.D.)--University of South Florida, 2006.
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by Joni Burnette Pirnot.
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RecognizableLanguagesDenedbyTwo-dimensionalShiftSp aces by JoniBurnettePirnot Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:NatasaJonoska,Ph.D. CommitteeChair:LiliaWoods,Ph.D. GregoryMcColm,Ph.D. StephenSuen,Ph.D. MasahicoSaito,Ph.D. Xiang-DongHou,Ph.D. Dateofapproval: October24,2006 Keywords:Recognizablepicturelanguages,Dotsystems, Two-dimensionalniteautomata,Transitivity,Doublyper iodic c r Copyright2006,JoniBurnettePirnot

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Dedication ForHolly,whooncetoldmetoneverwastetalent.

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TableofContents ListofFigures iii Abstract v Preface vi 1Introduction 1 1.1HistoryandOverview.............................. 1 1.2NotationandTerminology.......................... .5 1.3DotSystems...................................12 2FiniteAutomataRecognizingTwo-dimensionalShiftSpace s17 2.1RecognitionofShiftsofFiniteTypeby M F ( X ) ...............19 2.2RecognitionofSocShiftsby M F ( X ) .....................28 2.3ForbiddenandForcedStructuresin M F ( X ) and M F ( X ) ...........34 3Transitivity 39 3.1FactorLanguagesofDotSystemsDenedover Z = 2 Z ............39 3.2GraphRepresentations............................ .54 4Periodicity 64 4.1DoublyPeriodicPointsinTwo-dimensionalShiftSpaces ..........65 4.2Examples....................................69 5MonoidsandFollowerSets 85 5.1MonoidsforDotSystems............................ 86 5.2FollowerSetsandPredecessorSets.................. ....100 i

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Conclusion 106 References 108 AbouttheAuthor EndPage ii

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ListofFigures 1.1Enclosureoftwoblocks............................ .9 1.2Dierenttypesoftransitivity..................... ....11 1.3Shapeswithinminimalrectangles................... ....13 1.4Extendingablockupward........................... 14 1.5Extendinganallowedblock......................... .16 2.1SamplesetofallowedblocksforanREClanguage........ ......18 2.2Horizontalandvertical k -concatenations...................20 2.3Fullshiftasvertexshift(a)andedgeshift(b)........ ........24 2.4GraphrepresentingtheThree-dotSystem............. .....24 2.5Comparisonof q*r s and q r 0 *s ....................26 2.6Anexampleofablockpathwhere k =2...................27 2.7Graph M F ( X ) representingtheDiagonal-shiftSystem............28 2.8SetofveallowedblocksforsubshiftofExample2.2.2.. .........30 2.9Shiftspacewhereany b issurroundedentirelyby a 's............31 2.10In M hF ( X ) ,graphdiamondsoflength2areforbidden.............35 2.11HorizontalgraphtriangleofCorollary2.3.6........ ..........38 3.1TheWallpaperPattern............................. 41 3.2Theinductivestep...............................4 6 3.3Theprocessofllinguptheblock B .....................49 3.4Extensionofablockintoatriangle.................. ....51 3.5Possiblepositionsforintersectionofrightisosceles triangles.........52 3.6Full-squareSystem............................... 55 3.7One-columnextensionhavingevensum................ ....56 3.8One-columnextensionhavingoddsum................. ...57 iii

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3.9A4 3blockof1'smeets4 3blockof0's.................57 3.10Uniformtransitivityindirection( u;v )....................60 3.11Blocksandblockpathsover2 2states...................61 4.1Pointwithleastdoubleperiod(6 ; 4)......................65 4.2Vertexshiftsinone-dimensionalcase............... ......70 4.3Pointsofdoubleperiod(2 ; 1)and(1 ; 2)....................71 4.4Subgraphsrepresentingpointofdoubleperiod(2 ; 2).............72 4.5Stateamalgamation............................... 75 4.6Non-isomorphicgraphsrepresentingpointsofperiod(3 ; 3).........78 4.7Amalgamationoftwopairsofstates.................. ....78 4.8Subgraphshowingforcedlabels..................... ...79 4.9Amalgamationofthreepairsofstates................ .....80 4.10Amalgamationofstates1 ; 5,and9dictatesotherlabels...........80 4.11Multipleamalgamationssuggestedby1 = 5 = 9amalgamation.........81 4.12Representedpointshaveperiod(1 ;n )forall n 3..............82 4.13Representedpointshaveperiod( n;n )forall n 4..............83 5.1Horizontaltranslationsof S ..........................94 5.2Strictlysocshiftwithfollower-separatedgraph.... ...........101 5.3Graphofone-dimensionalstrictlysocshift......... ........103 5.4Strictlysocshift;graphnotfollowerseparated..... ..........104 5.5Graphofreducedsizerecognizing Y ......................105 iv

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RecognizableLanguagesDefinedbyTwo-dimensionalShiftS paces JoniBurnettePirnot ABSTRACT Therearenumerousconnectionsbetweenthetheoryofformal languagesandthat ofsymbolicdynamics.Ineach,thetransitionfromonedimen siontotwodimensions isaccompaniedbymuchdicultydueinlargeparttotheempti nessproblem,whichis relatedtothepresence(orlackthereof)ofperiodicpoints andisknowntobeundecidable. Here,wefocusontwo-dimensionallanguagesthathavethepr opertythatallblocksallowed bythelanguagecanbeextendedtoacongurationoftheplane satisfyingthestructure ofthelanguage;forsuchlanguagestheemptinessproblemis notanissue.Werstshow thatdotsystemsmaybeassociatedwithtwo-dimensionallan guageshavingthisproperty, sothatwemightemploytheselanguagesasvariedexamples.W enextdeneanewtype ofniteautomatonandwithit,atoolforrecognizingtwo-di mensional\strings"ofdata. Itisthenshownthattheseautomatacorrectlyrepresentthe socshiftspacesthatresult fromtheapplicationofblockmapstoshiftsofnitetype.Th ereafter,theseautomata areutilizedtoinvestigatepropertiesoftransitivityint hetwo-dimensionallanguagesthat theyrepresent.Morespecically,newdenitionsfordier enttypesoftwo-dimensional transitivityareadaptedfromtopologicaldynamicsandthe nillustratedthroughtheuse ofdotsystems.Theappearanceofperiodicpointsinthelang uagesrepresentedbythese automataisalsoexplored,withamainresultbeingthatthee xistenceofaperiodicpoint isguaranteedundercertainconditions.Finally,issuesof equivalenceareintroducedin thetwo-dimensionalsettingwithregardstoformallanguag es(syntacticmonoids)and symbolicdynamics(thefollowersetsofagraphrepresentin gasocshiftspace). v

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Preface Ithasbeensaidthatthesumofthepartsisgreaterthanthewh ole.Albeitdicultfor amathematiciantocondonesuchillogicalrhetoric,Imusta cknowledgethatinmycase Indtruthinthisstatement.Thedevelopmentofmycognitiv eskills,thisdissertation, andmyoutlookonmathematicsandlifeingeneralhavebeenmo tivated,encouraged,and sustainedbymanypeopleoverthecourseofmylife.Iamgrate fulfortheopportunityto thanksomeofthosepeoplehere,althoughafewwillneverhav etheopportunitytoread herewhatalastingimpressiontheyhaveleftonme. Fromanearlyage,mymother,father,andsisterswerekinden oughtotoleratemy bookishnessandtooverlookmylackofcommonsense.Mymothe rplantedaloveof mathematicsinmyyoungheartwhensheexplainedpercents,d ecimals,andfractionsto meinawaythatillustratedthebeautifulintercomplexityo fmathematicaltruths,and myfathertaughtmethatquietmomentsdeepinthoughtcouldb esomeofthemost valuabletimesinaperson'slife.DeeandLeahhavealwaysha dwordsofencouragement forme,HollystartedmeonanacademicjourneywithwordsIwi llneverforget,andAmy accompaniedmeonpartofthatjourneyasshestudiedourcult ure'smostmemorable words. WhenIwasayoungstudent,MikeFramersttaughtmehowtodos eriousmathematics,andwhenIthoughtIhadlostmyway,SooBongChaebeca memyguide.Itwas hewhopropelledmeontograduateschool.Onlyoneotherpers onwasasadamantabout myattendinggraduateschoolasSooBong,andthatwasJohn.I lookforwardtosharing thetitleofDr.Jwithhim. Whenmygraduate-schoollifecametoahaltinordertomakewa yfornewlife,my 103-year-oldfriendMuggietoldmethat\Lifeiswhathappen swhileyouarebusymaking plans".ZenaandMonawerebornin1990and1992,respectivel y,andtheyquicklybecame myraisond'^etre.Ithankthembothforthemanyimportantle ssonstheyhavetaughtme. vi

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WhenInallymademywaybacktoUSFafterave-yearhiatus,I metthewoman whowouldchangemylife.Dr.NatasaJonoskaassistedmeaca demicallyandemotionally, sheinspiredmemathematicallyandprofessionally,andshe supportedmepersonallyand nancially.(IamgratefulforfundingIreceivedthroughth egrantsCCF#0432009and EIA#0086015.)IwouldneverhaveattemptedenteringthePh. D.programwithouther guidance.Therearenotenoughwordstothankherproperly. Iwouldalsoliketothanktheothermembersofmycommitteewh odevotedtheirtimeto thereadingandeditingofthisdissertation.Inparticular ,Dr.GregoryMcColmhasbeen anintegralpartofmyeducation,andIthankhimforhisinput andinsight.Additionally,I thankDr.JarkkoKari,whoonseveraloccasionsoeredthoug htfuldialoguewithrespect tosymbolicdynamicsandautomatatheory. MyworkcouldnothavebeencompletedwithoutthesupportofM anateeCommunity College.Dr.JohnRosen,Dr.MikeMears,Dr.DennisRunde,an dAltay Ozgenerhave beenespeciallysupportiveandhelpfulthesepastseveraly ears. Finally,Iwishtothankmypsychiatrist,physician,barten der,banker,chaueur, masseuse,chef,andbestfriend.Thesumofthesepartsisone wholeperson-myhusband, StevePirnot-andIthankhimforthemany,manypartshehaspl ayedovertheyears. vii

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1Introduction Inthischapter,therequiredconceptsfromtheeldsofsymb olicdynamicsandformal languagetheoryareintroduced.Firstweoutlineahistoryo ftheresearchdoneinthese eldsinthetwo-dimensionalcase,andwegiveanoverviewof themainideasandresults foundwithinthispaper.Nextweprovidenotationandtermin ologyrequiredfordiscussion ofthetwo-dimensionalcase;inparticular,wedenesevera ltypesoftwo-dimensional transitivitybymodifyingsimilarnotionsfoundinthestud yoftopologicaldynamical systems.Finallyweexplaindotsystemsastheyexistinthel iterature,andwethenprove thatanyniteblockallowedbythestructureofadotsystema lsoappearsasasubblock ofsomecongurationoftheplanefoundwithinthatdotsyste m. 1.1HistoryandOverview Onewaytostudyachanging(dynamical)systemistomakethet imediscretesoasto studytheiteratesofasingleactiononthesystem.Theeldo fsymbolicdynamicstakes thisideaastepfurtherbyalsomakingthespacethatreprese ntsthesystemdiscrete. Thegeneralideaistouseanitesetofstatestorecordtheac tiononthespacebyrst assigningasymboltoeachstateandthenkeepingtrackofthe sestatesindiscretetime stepsbyrepresentationviaaninnitesequenceofsymbols[ 25].Soasymbolicdynamical systemstudiesaspaceofinnitesequencesbeingactedupon byatranslationmapthat shiftsobservationfromonepartofasequencetoanotherpar tofthesequence.Sucha systemisreferredtoasashiftspace,andtheinnitesequen cesarereferredtoaspointsof theshiftspace.Onewaytokeeptrackoftheactiononcertain shiftspacesistouseanite automaton,whichisadirectedgraphwithtransitionsbetwe enthestatesthatrepresent thediscretetimesteps.Thislinksthestudyofsymbolicdyn amicswiththatofformal languages,sincerecognizablelanguagesarepreciselytho sethatcanberepresentedbya niteautomaton. 1

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Whilemanywell-developedtheoriesexistinonedimensionf orthestudyofsymbolic dynamics[25],automatatheory[21,28],andformallanguag es[10,22,29],manyofthese resultsdonotholdtrueintwodimensions.Thetransitionfr omonetotwodimensions iscomplicatedbyseveralfactors,oneofwhichisrelatedto thepossiblenon-existenceof periodicpointsinatwo-dimensionalshiftspace.Asanillu stration,considerthefollowing questionposedbyHaoWang[31]in1961:Isitdecidablewheth eranitesetofequal-sized squaretileswithcolorsoneachedgecantiletheplaneinsuc hawaythatcontiguousedges willalwayshavethesamecolor?Wangansweredhisownquesti oninthearmative(see, forexample,[32])byconstructinganalgorithmthathinged ontheassumptionthatanyset oftilescapableoftilingtheplanewouldadmitaperiodicti ling.Atthetime,itseemeda reasonableassumption:thesetofallpossibletilingsofth eplaneusingagivensetofWang tilesdenesatwo-dimensionalshiftofnitetype,andinth elanguageofone-dimensional symbolicdynamics,ashiftofnitetypeisnonemptyifandon lyifitcontainsaperiodic point[25].However,ina1966publication,RobertBergersh owsthatWang'ssolutionis incorrect[2]bydemonstratingtheexistenceofasetofWang tilesthatcanonlytilethe planeaperiodically.Wang'squestionhascometobeknownas theemptinessproblem. Itisnowknown[7]thattheemptinessproblemisequivalentt othehaltingproblemfor Turingmachinesandisthereforeundecidable. Eortstoinvestigatetwo-dimensionalrecognizablelangu agesdosointhecontextof niterectangularpictures(arraysofsymbols)andnon-rec tangularshapes[1].In[7], GiammarresiandRestivointroducetheclassRECofrecogniz ablepicturelanguagesas thosethatcanbeobtainedbyprojectionofalocalpicturela nguage.(Alocalpicture languageoverthealphabetisdenedasonethatcanbecompl etelydescribedbya setofallowed2 2tilesover [f # g ,with#beinganon-alphabetsymbolplaced aroundtheborderofeachpicture.)Itisknownthattheclass RECisnotclosedunder complementation[8],whichmotivatesthediscussionofhie rarchywithinthefamilyoftwodimensionallanguages.Forexample,KariandMoore[18]sho wthatlanguagesrecognized by4-wayalternatingniteautomataareincomparabletoREC .Itisdemonstratedin [23]thateveryrecognizablepicturelanguagecanbeobtain edastheprojectionofanhvlocalpicturelanguage.(Inanhv-localpicturelanguage,t he2 2tilesthatdescribethe languagearereplacedbyhorizontalandverticaldominoes1 2tilesand2 1tiles, respectively-sothathorizontalandverticalreadingofth epicturescanbeaccomplished 2

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separately.)Thissuggestsrepresentationoftwo-dimensi onallanguagesthroughtheuseof twoseparategraphsormatrices[14,26].However,themaind rawbacktohavingseparate graphsforhorizontalandverticalmovementisthatwhenabl ockmapisappliedtothe graphrepresentingthelocallanguage,thenewly-labeledg raphfailstocorrespondto thesoclanguageintended[5]:theinherentpropertiesoft hesymbolsthatresultfrom interlacinghorizontalandverticalmovementcannotbedes cribed.Otherattemptsto representpicturelanguagesfocusontheuseofaparticular kindofcellularautomaton [11,13].Acomprehensivesurveyoftheresearchdoneintwodimensionalniteautomata duringthetimeperiodbeginningwithBlumandHewittin1967 [4]andupto1991can befoundin[12],whileanexcellentsurveyofmorerecentres ultscanbefoundin[18]. Thefocusofthisdissertationisontwo-dimensionalrecogn izablelanguageshaving thepropertythatanynitepictureallowedbythestructure ofthelanguagemaybe innitelyextendedtosomecongurationoftheplanethatal sosatisesthestructure ofthelanguage;thatis,wheretheemptinessproblemissolv ablesincethelanguageis prolongable .Furthermore,weshallbeinterestedintwo-dimensionalla nguagesthatare factorial ;thatis,languageshavingthepropertythatforanyblockfo undinthelanguage, allofitssubblocksarealsofoundinthelanguage.Inonedim ension,therearewelldevelopedtheoriesconcerninglanguagesthatarefactoria l,prolongable,andrecognizable (FPR-languages).Herewedevelopatheoryfortwo-dimensio nalfactorial,prolongable, andrecognizablelanguages(2DFPR-languages)whichareth efactorlanguagesofcertain two-dimensionalshiftspaces. Offurtherinterestinonedimensionisthesetoffactorial, transitive,andrecognizable languages(FTR-languages)thatareasubsetoftheFPR-lang uages.Forone-dimensional languages,thereisonlyonenotionof transitivity :givenanytwoblocksfoundinthelanguage,thereexistsathirdblock,alsointhelanguage,whic hcontainsthegivenblocks assubblocks.InSection1 : 2itisdemonstratedthatseveraldierentnotionsoftransi tivityexistfortwo-dimensionallanguagesandthateachde nesaninvariantpropertyfor two-dimensionalshiftspaces.\Dotsystems",whichareini tiatedin[19],aredescribedin Section1 : 3anditisthenshownthattheseshiftspacesbelongtothecla ssof2DFPRlanguages.Dotsystemscanthereforeproviderichexamples forthetheoryfoundinsubsequentchapters. InSection2 : 1,wedeneanewtypeofautomatonthatiscapableofrecogniz ing 3

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2DFPR-languages.Bydispensingwiththeboundarysymbolth atformstheborderof thepicturesfoundinthetwo-dimensionallanguagesbelong ingtotheclassREC,thisnew typeofconstructionallowstheautomatontogeneratetwo-d imensionalshiftspaces(and therefore,two-dimensionalfactorlanguages)inawayquit esimilartothewayinwhich one-dimensionalshiftspacesaregenerated.Itisexplaine dthatthecrucialcomponent oftheautomaton'sconstructionisadenitionofacceptanc ethatgivestheautomaton specicinstructionsregardingthedimensionsandstructu reofblocksthataredeemedto berecognizable.InSection2 : 2,itisveriedthatthisnewtypeofautomatoncorrectly representstheimageofashiftspace(ofnitetype)underab lockcode;nosuchgraphrepresentationwiththiscapabilityexistsintheliterature. ThroughoutChapter2,theclass ofdotsystemsisemployedtogeneratesomemanageableexamp lesoftwo-dimensional shiftspacesandtheirgraphrepresentations.Section2 : 3closesthechapterwithseveral PropositionsandCorollariesthatwillbeofuseinthediscu ssionofperiodicitythatis foundinChapter4. Chapter3revisitsthenotionofdierenttypesoftransitiv ityexistingintwo-dimensional languages.Mostoftheresultsontransitivityfoundwithin thischapterhavealreadybeen publishedin[16].InSection3 : 1,asanillustrationofthevarioustypesoftwo-dimensiona l transitivity,dotsystemsarepartiallycategorizedbased ontheshapesthatdenethem. InSection3 : 2,itisshownhowthetypeofgraphrepresentationdenedinC hapter2 canrevealinformationregardingtransitivityintherelat edfactorlanguages.Inparticular,amainresultofthechapterandofthedissertationisth atfora2DFPR-language, thereisanalgorithmthatdecideswhetherthegivenlanguag eexhibitsaparticulartype oftransitivity. InSection4 : 1,themainresultfromChapter3islinkedtotheexistenceof periodic pointsundercertainconditions.Section4 : 2thenoersdetailedexamplesofperiodic pointsfoundintwo-dimensionalshiftspaceswithrespectt otheappearanceofsuchpoints incorrespondinggraphrepresentations. Finally,issuesofequivalencefortheblocksofatwo-dimen sionallanguagearediscussed inChapter5.InSection5 : 1amonoidisdenedbasedonequivalenceclassesforthe blocksofapicturelanguage,andthensomepartialresultsa reachievedwithrespectto dotsystems.AdierentapproachissuggestedinSection5 : 2byinvestigatingthefollower setsofblocksthatactasthestatesofagraphrepresentinga two-dimensionalsocshift 4

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space.Thesenaltopicsserveasanintroductiontothemany openquestionsthatexist intheeldoftwo-dimensionalformallanguagetheorywithr egardstosymbolicdynamics andniteautomata. 1.2NotationandTerminology Fornotation,terminology,andbasicresultsofone-dimens ionalsymbolicdynamicalsystems,see[25].Fornotation,terminology,andbasicresult sofone-dimensionalformal languagetheory,see[10].Someadditionalnotationandter minologywillberequiredfor thediscussionofthetwo-dimensionalcase. Givenanitealphabet,denethe two-dimensionalfull -shift tobe Z 2 : A point x 2 Z 2 isafunction x : Z 2 ,thatis,acongurationoftheplanewheretheinteger lattice Z 2 hasbeenpopulatedwithchoicesfromthealphabet.For x 2 Z 2 and w 2 Z 2 wewillsometimesdenote x ( w )as x w andmayrefertothecoordinatepoint w 2 Z 2 asa cell.Similarly,for x 2 Z 2 and R Z 2 ,let x R denotetherestrictionof x to R .Wecall R aregion,andwecallaniteregion S Z 2 a shape .Inparticular,[ j;j ] 2 isthesquare shapeofsize2 j +1centeredattheorigin. Theset Z 2 isacompactmetricspaceunderthemetric ( x;y )=2 j ,wherefor x;y 2 Z 2 j isthelargestintegersuchthat x [ j;j ] 2 = y [ j;j ] 2 : (When x = y ,dene ( x;y )=0.)Informally,theclosertwopointsaretoeachother,th elargerthecentered squareshapeonwhichtheyagree.For v 2 Z 2 ; denethe two-dimensionaltranslationin direction v as v where v isdenedby( v ( x )) w = x w + v .Asubset X Z 2 issaidtobe translationinvariantifforall v 2 Z 2 v ( X ) X .If X Z 2 istranslationinvariantand closedwithrespecttothemetric ,wesaythat X isa two-dimensionalshiftspace (ora subshift ofthefullshift). Wedenea design r onashape S tobeafunction r : S ,wherethegivenshape S hasbeen normalized sothatmin f i :( i;j ) 2 S g =0andmin f j :( i;j ) 2 S g =0.In otherwords,theshapehasonlynon-negativeintegercoordi nateswithboundarieslying onthecoordinateaxes.Thenumberofoccurrencesofthesymb ol a 2 inadesign r 5

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shallbedenoted j r j a .Ifisasetofdesignsonaxedshape S; thentheset X := f x 2 Z 2 : 8 v 2 Z 2 ; ( v ( x )) S 2 g (1.2.1) isatwo-dimensionalshiftspacethatiscalleda two-dimensionalshiftofnitetype .For shiftsofnitetypedenedthroughanitesetofseveraldi erentshapes,thereisnoloss ofgeneralityinassumingthat X isdenedthroughasinglerectangularshapehavingsize sucienttocontainallothershapes. Givenadesign r onarectangularshape T Z 2 having m rowsand n columns,we call r an m n block anddenotesuchdesignsby B m;n .Foreaseofnotation,wemay sometimesdropthesubscriptswhenthenumberofrowsandcol umnsisirrelevant,and wemayrefertoblocksas i whentheindexoftheblockdoesnotrefertoitsdimension. Weshallsaythatan m n blockhas height m length n ,and thickness k =max f m;n g : If m =0or n =0,then B m;n isthe emptyblock andisdenotedby .Foradesign B : T ,a subblock B 0 of B istherestrictionofthedesigntoarectangularsubset T 0 T Z 2 .Insuchcases,wesometimessaythat B encloses B 0 anddenotethisby B 0 B .Forxed r and c ,thesetofall r c subblocksof B isdenotedas F r;c ( B ),and thesetofallrectangularsubblocksof B isdenotedwith F ( B ).Thesetofallblocksof axedsize m n overisdenoted m;n ,andthesetofallblocksofanysizeoveris denotedby Alanguage L isanysubsetofafreemonoid.(Amonoidisasetwithabinary associativeoperationandanidentity.)A picturelanguageover thealphabetisdened tobeasubsetof : Inparticular,a localpicturelanguage L isonewhere B 2 L ifandonlyif F k;k ( B ) Q ,where Q isanitesetofallowed k k blocks.Alsoof interestwillbethoselanguagesthatare factorial :alanguage L issaidtobefactoriali L = F ( L ):= f F ( B ) j B 2 L g .Alllanguagesunderdiscussioninthispaperare recognizable languages:thatis,languagesthatcanberepresentedbyan iteautomaton.Inparticular, GiammarresiandRestivo[7]introducetheclassRECof recognizablepicturelanguages as thosethatcanbeobtainedbytheprojectionofalocalpictur elanguage,wherethelocal languageistakenoverthealphabetandtheset Q ofallowed2 2tilesistakenover [f # g ,with#beinganon-alphabetsymbolplacedaroundtheborder ofeachrectangular picture. 6

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Thereisalanguageassociatedtoeachshiftspace.Wesayabl ock B : T occurs in X Z 2 ifthereexistsan x 2 X suchthat x T = B .The factorlanguage ofashift space X is F ( X ):= f F m;n ( x ): m;n 0 ;x 2 X g ; (1.2.2) i.e.thecollectionofallsubblocksthatoccurinpointsof X .Forshiftspaces,thefactor languageoftheshiftspaceuniquelydeterminestheshiftsp ace;thatis,fortwoshift spaces X and Y X = Y ifandonlyif F ( X )= F ( Y )[25].Forthisreason,whenagraph representsthefactorlanguageofashiftspace,weshallmor egenerallyrefertothegraph asarepresentationof X Let X beatwo-dimensionalshiftofnitetypedenedbyasetofdes ignsona normalizedshape S .Fortheshape S ,denethenumberofrowsin S tobe r =1+max f j : ( i;j ) 2 S g anddenethenumberofcolumnsin S tobe c =1+max f i :( i;j ) 2 S g : We shallrefertoashape S ashavingdimension r c although S maybeapropersubsetof thecellsthatcomprisethenormalized r c rectangle T .Cellsthatappearin T butnot in S willbeofparticularinterest. Denition1.2.1 Givenan r c shape S andthe r c rectangle T thatcontainsit, w 2 T iscalleda freecell if w= 2 S Forthickness k =max f r;c g ,set Q = F k;k ( X )andlet beanormalized k k square shape.Thereisnolossofgeneralityinassumingthattheshi ftofnitetype X isdened by Q ratherthanby,thatis, X := f x 2 Z 2 : 8 v 2 Z 2 ; v ( x ) 2 Q g : (1.2.3) Notealsothatif X isatwo-dimensionalshiftofnitetypedenedthroughaset ofblocks F k;k ( X ),thenforall K k X mayalsobedenedthrough F K;K ( X ).Insomecases, however,itwillbepreferabletoemploythesetofblocks F r;c ( X )=whosedimensions minimallycontaintheshape S .Inparticular,the allowedblocks ofashiftofnitetype X isdenedhereasthelocalpicturelanguage A ( X )ofallblocks B 2 thatsatisfythe condition F r;c ( B ) .(For B = B m;n with m
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shiftspace,weshalloftenusetheset F r;c ( X )=inordertosimplifytheproofs,but whendiscussingthesymbolicdynamicsofatwo-dimensional shiftspace,weshallndit preferabletoemploytheset F k;k ( X )= Q .Wepointoutthatforthesetofblocks F ( X ), F r;c ( B ) isnecessaryfor B 2 F ( X )butisnotsucient: B mustalsooccurinsome pointof X .Ontheotherhand,forthesetofblocks A ( X ), F r;c ( B ) isbothnecessary andsucientfor B 2 A ( X ). Foraone-dimensionalshiftofnitetype,thefactorlangua geoftheshiftspaceisalways alocallanguage;thatis, A ( X )= F ( X )forallone-dimensionalshiftsofnitetype[25]. Inatwo-dimensionalshiftofnitetype,however,ablockin A ( X )neednotappearasa blockin F ( X )sincetheemptinessproblemraisesthequestionofwhether thelocalpicture language A ( X )isprolongable.Forexample,Kari[17]providesasmallape riodicsetof Wangtilesdescribingashiftofnitetype X havingthepropertythat F ( X ) ( A ( X ): usingthegivensetofWangtiles,onecanconstructblocksth atconformtothestructure ofthelanguageyetcannotbeextendedanyfartherincertain directionsandtherefore cannotappearintheset F ( X ). JustasRECreferstotheclassoflanguagesthatcanbeobtain edthroughtheprojection oflocalpicturelanguages,two-dimensionalshiftsofnit etypemaybe\projected"toform anotherclassofshiftspaces.Morespecically,foragiven two-dimensionalshiftofnite type X ,wecantransform x 2 X intoanewpoint y 2 Y where Y Z 2 employssome newalphabet.Forthe k k squareshape T ,afunction: F k;k ( X ) thatmaps k k blocksin X tosymbolsinby( x T + w )= y w iscalleda k k -blockmap .The map : X Z 2 denedby y = ( x )with y w inducedbyiscalleda k k -blockcode anditsimage Y = ( X )iscalleda two-dimensionalsocshift .Whentheblockcode is invertible,wereferto asaconjugacyandsaythatthespaces X and Y are conjugate Akeyfeatureofblockcodesisthatforablockcode : X Y andapoint x 2 X computing attheshiftedpoint ( i;j ) ( x )givesthesameresultasshiftingtheimage ( x ) using ( i;j ) inthespace Y .Thatis,thediagramin(1.2.4)commutes([25].) Insymbolicdynamics,weareofteninterestedinproperties thatare invariant ;that is,propertiesthatholdtrueforallshiftsthatareconjuga tetoagivenshift.Fortwodimensionallanguages,wewillbeinterestedinwhetheragi venpairofblocksmight coexistwithinasinglepointoftheshiftspace.Inonedimen sion,suchaquestionis oneoftransitivity.Unliketheone-dimensionalcase,howe ver,thereareseveraltypesof 8

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transitivitythatappearintwo-dimensionallanguages,ea chofwhichdenesaninvariant propertyforconjugateshiftspaces(seeProposition1.2.4 foraproof). ( i;j ) X X ## Y Y ( i;j ) (1.2.4) Todiscusstransitivityinthetwo-dimensionalcase,wers tneedtodenedistanceand directionbetweenapairofblocksinatwo-dimensionalspac e. Denition1.2.2 Ablock B enclosesthepairofblocks B 0 and B 00 if B 0 ;B 00 2 F ( B ).Furthermore,ablock B minimallyencloses B 0 and B 00 if B encloses B 0 and B 00 insuchaway thatboththebottomandtoprowsof B aswellastheleftandrightcolumnsof B all intersectatleastoneoftheblocks B 0 ;B 00 Ifablock B m;n minimallyenclosesthepairofblocks B 0 p;q B 00 s;t wesaythat d ( B 0 ;B 00 )= max f 0 ;m p s;n q t g isthe distanceatwhich B 0 meets B 00 (u',v') B B '' B d q p t s (u'',v'') Figure1.1:Enclosureoftwoblocks Let L beatwo-dimensionallanguagecontainingtheblocks B 0 and B 00 .Ifthereisa block B in L thatminimallyencloses B 0 and B 00 withoutallowingthemtooverlap,where thebottom-leftcornersof B 0 and B 00 appearatvertices( u 0 ;v 0 )and( u 00 ;v 00 )respectively,we saythat B 0 meets B 00 within L indirection ( u;v ) forany( u;v )havingintegercoordinates thatisanon-zeromultipleof(i.e.,parallelto)thevector ( u 00 u 0 ;v 00 v 0 ). 9

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Informally,wesaythat B encloses B 0 and B 00 ifboth B 0 and B 00 appearassubblocks of B .ThisisdepictedinFigure1.1,wheretheblockthatminimal lyencloses B 0 and B 00 is indicatedwithdottedlines.Thedirection( u;v )inwhich B 0 and B 00 meetisdetermined bythebottom-leftcornersof B 0 and B 00 ,andtherefore u;v 2 Z .Notethatitmightbe thecasethatthetwoblockstouch,inwhichcasethedistance atwhichtheymeetwould be0.Denition1.2.3 Wesaythatatwo-dimensionallanguage L is transitiveindirection ( u;v ) ifforeverypairofblocks B 0 ;B 00 2 L theblock B 0 meets B 00 indirection( u;v )within L transitive ifforeverypairofblocks B 0 ;B 00 2 L thereisablock B 2 L thatencloses B 0 and B 00 uniformlytransitive ifthereisapositiveinteger K suchthatforeverypairofblocks B 0 ;B 00 2 L thereisablock B 2 L thatminimallyencloses B 0 and B 00 inawaythat d ( B 0 ;B 00 ) K uniformlymixing ifthereisapositiveinteger K suchthatforeverypairofblocks B 0 ;B 00 2 L ,theblock B 0 meets B 00 ineverydirectionwithin L provided d ( B 0 ;B 00 ) > K DierenttypesoftransitivityarepresentedinFigure1.2. Directionaltransitivityis worthnamingintwoparticularcases:If L istransitiveindirection(1 ; 0),weshallsaythat L is horizontallytransitive (seediagram(a)inFigure1.2),andsimilarlywesaythat L is verticallytransitive if L istransitiveindirection(0 ; 1).Transitivityismoregeneralthan uniformtransitivityasthereisnoboundonhowfarapartthe blocks B 0 and B 00 mightbe: thatis,uniformtransitivityensuresthatblocksneednote xtendtoofarinordertomeet eachother(seediagram(b)inFigure1.2).Anotionof uniformdirectionaltransitivity couldbedenedinthesamemanner.Mixingallowsfortwobloc kstomeeteverywhere outsideofacertain\neighborhood",whereasuniformmixin gguaranteesthatthereis aboundonthe\radius"ofthisneighborhoodregardlessofth esizeoftheblocks.For 10

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exampleindiagram(c)ofFigure1.2,theshadedregionin(c) indicatesthe\neighborhood" denedbytheconstant K ;here,theblock B 00 canappearinanydirectionfrom B 0 provided itisoutsidethisneighborhood.Fromthedenitionspropos edhere,onecanseethatwe havethefollowingimplications: mixing )8 ( u;v ) ; transitiveindirection( u;v ) ) transitive(1.2.5) Furthermore,theimplicationsin(1.2.5)stillholdtruewh enweinserttheword\uniform" infrontofeachproperty. (b) (c) B' B'' B B' B'' B B'' (a) < K K B'' B' Figure1.2:Dierenttypesoftransitivity Inthecaseofone-dimensionalrecognizablelanguages,all notionsof(uniform)transitivityandmixingcoincide.Directionaltransitivitymo stcloselyresemblesthenotion ofone-dimensionaltransitivity,asonecanexamineabi-in nitesequenceof\blocks"in thespecieddirection.Thedenitionspresentedherefort ransitivityandmixingintwodimensionallanguagesaresimilartothosedenedfortopol ogicaltransformationgroups [27,9]inthestudyoftopologicaldynamics.Proposition1. 2.4describestransitivityin thetwo-dimensionalcaseasaninvariantpropertyforconju gateshiftspaces.Proofsfor theinvarianceofmixingand/oruniformitypropertiescanb eaccomplishedinasimilar fashion.Proposition1.2.4 Let X and Y betwo-dimensionalshiftspaces,andlet : X Y be aconjugacyfrom X to Y .If L = F ( X ) istransitive,then L 0 = F ( Y ) isalsotransitive. Proof. Let B 0 : T 0 and B 00 : T 00 beblocksinthelanguage F ( Y )overthe alphabet.Weseekablock B 2 F ( Y )thatencloses B 0 and B 00 .Since isinvertible, B 0 and B 00 haveuniquepreimages,say 1 ( B 0 )= 0 and 1 ( B 00 )= 00 .Furthermore, 11

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since F ( X )istransitive,thereexistsablock 2 F ( X )thatencloses 0 and 00 .Thatis, theremustexist w 0 ;w 00 2 Z 2 suchthat ( T 0 + w 0 )= 0 and ( T 00 + w 00 )= 00 .Therefore, ( )= B 2 F ( Y )isthedesiredblocksince B ( ( T 0 + w 0 ))= ( 0 )= ( 1 ( B 0 ))= B 0 and B ( ( T 00 + w 00 ))= ( 00 )= ( 1 ( B 00 ))= B 00 1.3DotSystems When G isanitegroup, G Z 2 isagroupviacoordinate-wiseproduct.Let X bea two-dimensionalshiftspacewhichisalsoasubgroupof G Z 2 :Suchasubshiftiscalleda two-dimensionalgroupshift .Ithasbeenshown[19]thatalltwo-dimensionalgroupshift s areshiftsofnitetype.Inapublicationdated1992,Kitche nsandSchmidtdeneatype ofgroupshiftthattheycalladotsystemandthenshowthatev erytwo-dimensionalgroup shiftisaniteintersectionofthesedotsystems[20,30].F ortheworkherein,weshall employdotsystemsthataredenedviasomeshape S overthegroup G = Z = 2 Z .Thatis, X := ( x 2f 0 ; 1 g Z 2 : 8 v 2 Z 2 ; X w 2 S + v ( x w )=0 ) : (1.3.6) Informally,apoint x belongstothistypeofdotsystem X ifandonlyifalltranslatesof theshape S in x containanevennumberof1 0 s .Dotsystemsmayalsobedenedover niteabeliangroupsotherthan Z = 2 Z intheobviousway:thatis,theproductofthe symbolsthatappearwithintranslatesof S shouldequatetotheidentityelementforthat group.(SeeLemma1.3.1andProposition1.3.4tofollow.)It isusefultothinkof S asa shapeofdotswithinadeningrectangle T asillustratedinFigure1.3.Inparticular,the subshiftdenedthroughshape(a)isknownintheliterature astheThree-dotSystem.For convenience,weshallalsooccasionallyrefertotheelemen tsof S simplyasdots.Unless statedotherwise,wedisregardthesingletoncasewhen S =(0 ; 0),astheshiftspace resultingfromthisshapeisthetrivialonecomprisedofthe singlepointofallzeros. Dotsystemsshareaparticularlynicepropertythatallowsu stoemploythesetof allowedblockswheninvestigatingpropertiesofthefactor languageoftheshiftspace:that is,anyblockallowedbythestructureofthedotsystemalsoa ppearsasasubblockofsome 12

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(a)(b) (c) (d) (e) Figure1.3:Shapeswithinminimalrectangles pointintheshiftspace.Theproofofsuchisfacilitatedbyt heobservationthatindot systems,anyblockthatistoonarrowtocontaintheshape S maybe\extended"intoa larger(allowed)blockthroughtheassignmentofadditiona lrowsand/orcolumns. Lemma1.3.1 (BlockExtension) Foradotsystem X denedbytheniteabeliangroup G overan r c shape S f G m;n : m>><>>>: B 0 m;n ( i;j )for0 i n 1 ; 0 j m 1 e for i n or j m; ( i;j ) 6 =( i 0 ;j 0 ) ( Q ( s;t ) 2 S \ T 0 B 0 m;n ( s;t )) 1 for( i;j )=( i 0 ;j 0 ) (1.3.7) (Theassignmentoftheidentityelementin(1.3.7)ischosen onlytoeasenotation:one shouldnotethatwiththeexceptionofthexeddot,allassig nmentsarearbitrary.) Thecasewhenonlyoneof m or n issmallerthan r or c ,respectively,istreated similarly;however,careshouldbetakeninthechoiceofwhi chdotfrom S toxinitially. Withoutlossofgenerality,supposethat B 0 m;n : T 0 G issuchthat m
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elementischosenmerelytoeasenotation.) B ( i;j )= 8>>>>><>>>>>: B 0 m;n ( i;j )for0 i n 1 ; 0 j m 1 e for0 i n 1 ;m j
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Corollary1.3.2 Foradotsystem X denedbythegroup G = Z = 2 Z oversome r c shape S j F m;n ( X ) j =2 mn whenever mc .SimilartotheproofofLemma 1.3.1,onecanbegintodeneablock B 2 F k;k ( X )byxingadotinthetoprowofthe normalizedshape S beforearbitrarilypopulatingallothercellsofthenormal ized r c rectangularshape T 0 .Thenforeachof r c horizontaltranslatesof S ,theshape S remains enclosedbythe k k squareshape T ,sothateachcellintheemptycolumnintersecting thenewly-translatedshape S maybearbitrarilypopulated-withtheexceptionoftheone cellcontainingthetranslateofthexeddotwhosevalueisd ictatedbytherequiredsum. After r c horizontaltranslatesoftheshape S ,theshapeisnolongerenclosedby T ,so thatanyremainingcellsof T maybearbitrarilypopulated.Therefore,forthe k 2 cells locatedinshape T ,allbut1+ j r c j maybearbitrarilypopulatedwitheither0or1,and theresultfollows.When c>r ,theproofisanalogous. Proposition1.3.4istruefordotsystemsdenedoveranyni teabeliangroup G Proposition1.3.4 Forthedotsystem X denedovertheniteabeliangroup G A ( X )= F ( X ) Proof. Sincealldotsystemsareshiftsofnitetype, X adotsystem ) X isshiftofnite type ) F ( X ) A ( X ). Forthereverseinclusion,suppose B 0 m;n : T 0 G issuchthat B 0 m;n 2 A ( X ).Wewill demonstratethatthereexistsablock B m +2 ;n +2 2 A ( X )suchthat B m +2 ;n +2 ( T 0 +(1 ; 1))= B 0 m;n (seeFigure1.5).ThenbyKonig'sLemma, B 0 m;n mustoccurinapointoftheshift space X ,sincethereexistsaninniteprocessbywhichonecanexten dtheblockvia 15

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selectionfromanitesetofacceptablecellassignments.W ithoutlossofgenerality,we mayassumethat B 0 m;n issuchthat m r and n c .(Otherwise,usetheappropriate blockextensionfromLemma1.3.1toconstructsomelargerbl ockin A ( X )thatencloses B 0 m;n asasubblock.)Considerthesubblock 1 = f B 0 m;n ( i;j ):0 i n 1 ;m ( r 1) j m 1 g .Then 1 hasdimension( r 1) n ,andbyLemma1.3.1,thereexistsan upwardextensionof 1 .Notethattranslatesof S thatlieentirelyin B 0 m;n arenotaected bytheone-rowextensionof 1 :therefore,thisextensionof 1 alsoservesasanextension of B 0 m;n .Denotetheextendedblockas B 00 m +1 ;n 2 A ( X ),andnextconsiderthesubblock 2 = f B 00 m +1 ;n ( i;j ): n ( c 1) i n 1 ; 0 j m g havingdimension( c 1) ( m +1). ByLemma1.3.1,thereexistsanextensiontotherightof 2 resultingintheconstruction 4 1 2 3 Figure1.5:Extendinganallowedblock ofanewblock B (3) m +1 ;n +1 2 A ( X ).However,tomaintainconstructioninaclockwise direction,weuse j 0 =min f j :( c 1 ;j ) 2 S g toxthedot( c 1 ; ( m +1) r + j 0 ) forthetranslate S +(0 ; ( m +1) r ).Wethenassigncellvaluesfor 2 ,beginningat thetopofthenewcolumnandworkingourwaydownward.Tocont inuetheprocess, considerthesubblock 3 = f B (3) m +1 ;n +1 ( i;j ):0 i n; 0 j r 2 g havingdimension ( r 1) ( n +1).Wemayextend 3 belowbyrsttaking i 00 =min f i :( i; 0) 2 S g and xingthedot(( n +1) c + i 00 ; 0)inthetranslate S +(( n +1) c; 0).Wethenbegin toassignalargerblock B (4) insuchawaythat B (4) m +2 ;n +1 ( i;j )= B (3) m +1 ;n +1 ( i;j 1)for 0 i n; 1 j m +1,andcompletetheassignmentbyextending 3 (andhence B (4) m +2 ;n +1 )bysequentiallyinspectingtranslatesof S totheleftoftheinitialxeddot. Finally,usingthesubblock 4 = f B (4) m +2 ;n +1 ( i;j ):0 i c 2 ; 0 j m +1 g B (4) m +2 ;n +1 maybeextendedtotheleftbyxingadot(0 ;j 00 ) 2 S for j 00 =max f j :(0 ;j ) 2 S g inspectingtranslatesof S upwardfromthisdot,andsoon,toform B m +2 ;n +2 16

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2FiniteAutomataRecognizingTwo-dimensionalShiftSpace s Thegraphconstructiondenedinthischapterisbasedonint erlacinghorizontaland verticalmovementinthegraphs'transitions.Inthisway,t hegraphrepresentationtakes fulladvantageoftherichcomplexityofthoselanguagestha tarethefactorlanguagesof two-dimensionalshiftsofnitetypehavingthepropertyth at A ( X )= F ( X ).Themain componentoftheconstructionisatoolforrecognizingrect angularblocksthatconformto thestructureofthelanguage.InSection2 : 2itisveriedthatforatwo-dimensionalshift ofnitetypehavingproperty A ( X )= F ( X ),theconstructedgraphsaccuratelyrepresent theshiftspacesthatresultfromapplyingablockcodetothe shiftspace X .Intheproof ofProposition2.3.3,anexplicitexampledemonstrateshow toapplyahigherblockcode tothestates F k;k ( X )= Q ,whichchangesthealphabetbutnotthedynamics,sothat theresultingshiftspaceisconjugatetotheoriginalbutha sstatesofsize2 2.Thisin turnallowsustomakeseveralobservationsaboutcertainst ructureswithinthegraphand aboutthelocalizedrecognizablepicturesthattheyforce. WhenGiammarresiandRestivodenetheclassRECofrecogniz ablepicturelanguages, theydosointhecontextofniterectangularpicturesthatc anbesurroundedbyanonalphabetbordersymbol.Bygoingtoahigherblockcodeasnee ded,onemayassumethat anylocallanguageinRECcanbedenedthroughanitesetofa llowed2 2blocksthat havebeenpopulatedbysymbolsfromthe(new)alphabet 0 andthenon-alphabetsymbol #.Fortwo-dimensionalshiftspaces,thereisasimilarnoti onofthelanguageofashiftof nitetypebeingcontainedwithinalocallanguagedenedby anitesetof2 2blocks. Thatis,foranyshiftspace X ,thereisalocallanguage A ( X )suchthat F ( X ) A ( X ), withthefactorlanguagesoftheshiftspacebeinglocalinth ecasewhen A ( X )= F ( X ). Ifalanguage L isinRECbutisnotlocal,then L mustbetheprojectionofsomelocal languagethatisinREC.Inthesamemanner,ifasocshiftspa ce Y istheimageundera blockcodeofashiftofnitetype X havingthepropertythat A ( X )= F ( X ),then F ( Y ) 17

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isinRECevenif Y isnotashiftofnitetype.(When Y isasocshiftspacethatisnot ashiftofnitetype, F ( Y )isnotalocallanguage). Thefollowingillustratessubtledistinctionsthatmustbe madeforatwo-dimensional picturelanguagewhereaboundarysymbolisinherentinthed enitionofthelanguage. Example2.0.5 Dene L f a;b g tobeatwo-dimensionalpicturelanguagewith L 2 REC suchthatforeveryblock B 2 L ,anyappearanceof b iscompletelysurroundedby a 's.Thelanguage L canbedenedbythesetofallowedblocksdepictedinFigure2 .1. Forexample,theblock B = ###### aab # # aaa # # baa # ##### (2.0.1) wouldnotbeinthelanguage,although B 0 = ######## aaaaa # # aaaba # # aaaaa # # abaaa # # aaaaa # ####### (2.0.2) wouldbeinthelanguage. # ## a a ba a a ab a b aa a a aa b # aa # # #a a # #a # a ## a # a# # a ## # a a# # a aa a Figure2.1:SamplesetofallowedblocksforanREClanguage Blocks B and B 0 in(2.0.1)and(2.0.2),respectively,indicatethat F ( L ) 6 = L .Therefore, thelanguage L ofExample2.0.5isnotafactoriallanguage.Asweconsidert hefactor 18

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languagesofshiftsspaces,alllanguagesinthesequelaref actorial.InSection2 : 2,weshall revisitExample2.0.5inthecontextoftwo-dimensionalshi ftspaces. 2.1RecognitionofShiftsofFiniteTypeby M F ( X ) Ourgoalistoconstructaniteautomatonthatwillallowthe inputdatatoconsist of m n blocksthatcanbescannedlocally(andintermittently)byb othhorizontaland verticaltransitions.Todosowouldrequiredistinctsetso fedgesforhorizontalandvertical transitions,say E h and E v ,respectively,andwouldrequirethatwedenewhatwemean byacceptance.Thegeneralideaisthatgivenaninputblock, wewillconsidersequences ofsymbolsthatappearinawindowofxedsizeaswescanthein putblockfromthe lower-leftcornertotheupper-rightcornerbytravelingin twodirections-upand/orto theright-withintheconstraintsoftheblock'sdimensions .Iftheautomatonacceptsall suchsequencesofsymbolsappearingasaresultofsuchmoves anditisdeterminedthat thesesequencesoverlapprogressivelyinsomesense(tobem adeclearlater),thenweshall saythattheblockitselfisacceptedbytheautomaton. Toeasetheformaldiscussionofconstructingniteautomat acapableofrecognizing two-dimensionalsubshifts,weshallrefertotheextension ofadesign(notnecessarily ablock)byonerow(column)oflength(height) k asa k -concatenation .Informally, beginningwitha k k block,weallowasequenceofconcatenationsconsistingof k 1 blocksconcatenatedhorizontallytotherightoftheuppermostsymbolsintheexisting designand/or1 k blocksconcatenatedverticallyabovetheright-mostsymbo lsinthe existingdesign.Moreformally,supposewebeginwitha k k blockandproceedtoextend thisblockbythevertical k -concatenationof m 0 -many1 k blocks.Let m = k + m 0 and let B m;k betheresultoftheseconcatenations.Wenextallowa k 1block B 0 k; 1 tobe horizontally k -concatenatedtotheexistingblock.Thebinaryoperationi sdenotedby andtheresultingdesign B m;k *B 0 k; 1 (seetherstdiagraminFigure2 : 2)isdenedby r ( i;j )= 8<: B m;k ( i;j )for0 i k 1 ; 0 j m 1 B 0 k; 1 (0 ;j ( m k ))for m k j m 1 : (2.1.3) Inthesameway,a k n block B k;n with n k ,maybeextendedbyvertical k concatenationwitha1 k block B 00 1 ;k .Thebinaryoperationisdenotedby ,andthe 19

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resultingdesign B k;n B 00 1 ;k isdenedby r ( i;j )= 8<: B k;n ( i;j )for0 i n 1 ; 0 j k 1 B 00 1 ;k ( i ( n k ) ; 0)for n k i n 1 : (2.1.4) Finally, k -concatenationontoashapeotherthanarectangleisallowe dprovidedthatthe shapeisgeometricallycongruenttoashapethatresultedfr omanitesequenceof k concatenations.Wecallsuchasequenceofallowed k -concatenationsa k -phrase ,resulting inasetofprogressivelyoverlappingblocksofsize k k .Weshalldenotea k k block B thatoccursinthe k -phrase P by B k;k P .Witheach k -phrase P ,wemayalsoassociate twofunctions s and t :each k -phrasestartswith s ( P )= ,where isa k k block; andeach k -phraseterminatesin t ( P )= ,where isalsoa k k block.Anexample ofanunderlyingshapefora k -phraseisdepictedinFigure2.2totheright. k k k k b b a w Figure2.2:Horizontalandvertical k -concatenations Nowsupposeatwo-dimensionalshiftofnitetype X isdenedby Q = F k;k ( X )for some k 2.If F ( X )= A ( X )sothatthefactorlanguageoftheshiftspaceislocal,then the niteautomaton M F ( X ) =( Q;E;s;t; )denedthrough Q isanitedirectedgraph obtainedasfollows.(Notethatifashiftofnitetypeiscom pletelydescribedbyasetof 1 1blocksthatdenethelocallanguage A ( X )= F ( X ),thentheshiftspacemustbe thefullshift,sincethiswouldimplythatanytwoalphabets ymbolsmaybeplacednext toeachother.Insuchcases,thefullshift X overthealphabetmayalsobedened throughtheset F 2 ; 2 ( X )= Q having j Q j = j j 4 .)First,denethevertexsetof M F ( X ) to be Q .Forexample,say q = q (0 ;k 1) :::q ( k 1 ;k 1) ... . ... q (0 ; 0) :::q ( k 1 ; 0) and r = r (0 ;k 1) :::r ( k 1 ;k 1) ... . ... r (0 ; 0) :::r ( k 1 ; 0) 20

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aretwoverticesin M F ( X ) .Next,theedgesetrepresentingthetransitionsbetweenth e statesof M F ( X ) isdenedtoconsistofhorizontalandverticaltransitions E h and E v respectively,suchthat E = E h [ E v and E h \ E v = ; .Forhorizontaltransitions,anedge fromstate q tostate r isdenedas e h 2 E h ifandonlyif q (1 ;k 1) :::q ( k 1 ;k 1) ... . ... q (1 ; 0) :::q ( k 1 ; 0) = r (0 ;k 1) :::r ( k 2 ;k 1) ... . ... r (0 ; 0) :::r ( k 2 ; 0) and q (0 ;k 1) :::q ( k 1 ;k 1) r ( k 1 ;k 1) ... . ... ... q (0 ; 0) :::q ( k 1 ; 0) r ( k 1 ; 0) = q (0 ;k 1) r (0 ;k 1) :::r ( k 1 ;k 1) ... ... . ... q (0 ; 0) r (0 ; 0) :::r ( k 1 ; 0) 2 F ( X ) : Inthiscase,thehorizontaledgeisdenoted e h = q*r andisgiventhelabelofthe k ( k +1)blockthatistheresultofthehorizontal k -concatenationof q with r := f r ( i;j ): i = k 1 ; 0 j k 1 g .The k ( k +1)block q* r shallbedenoted ( e h ). Similarlyforverticaltransitions,anedgefromstate q tostate r isdenedas e v 2 E v ifandonlyif q (0 ;k 1) :::q ( k 1 ;k 1) ... . ... q (0 ; 1) :::q ( k 1 ; 1) = r (0 ;k 2) :::r ( k 1 ;k 2) ... . ... r (0 ; 0) :::r ( k 1 ; 0) and r (0 ;k 1) :::r ( k 1 ;k 1) q (0 ;k 1) :::q ( k 1 ;k 1) ... . ... q (0 ; 0) :::q ( k 1 ; 0) = r (0 ;k 1) :::r ( k 1 ;k 1) ... . ... r (0 ; 0) :::r ( k 1 ; 0) q (0 ; 0) :::q ( k 1 ; 0) 2 F ( X ) : Heretheverticaledgeisdenoted e v = q r andisgiventhelabelofthe( k +1) k blockthat istheresultofthevertical k -concatenationof q with b r := f r ( i;j ):0 i k 1 ;j = k 1 g whilethe( k +1) k block q b r isdenoted ( e v ).Inadditiontothelabelingfunction alreadydescribed,toeachgraphwemayassociatetwootherf unctions s and t :Each edge e 2 E hasthe source atavertexdenotedby s ( e ) 2 Q ,andeachedgehas target ata vertexdenoted t ( e ) 2 Q .(Thecasewhere t ( e )= s ( e )ispermissible.) Tobeginthediscussionofthelanguagerecognizedby M F ( X ) ,wemustmakemore precisethemeaningofapathanditslabel.Deneapathin M F ( X ) tobeasequence 21

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= q 0 x 1 q 1 x 2 :::q p ofverticesandtransitions,wherefor0 i p q i 2 Q and x i 2f *; g aresuchthat s ()= q 0 t ()= q p ,and ( q i 1 x i q i ) 2 F ( X )forall i 2f 1 ; 2 ;:::;p g : The lengthofapathisdenotedby j j ,thenumberofverticesvisitedafterleaving q 0 .When iscomprisedsolelyofedgesfrom E h (i.e. x i = forall i 2f 1 ; 2 ;:::;p g ),weshall sometimesrefertoasan h -path ,andextendthisnomenclatureinthenaturalwayto v -paths, h -cycles, v -cycles ,andinparticular, h -loops and v -loops .Thelabelofapathshall bedenedinductively:For j j =1, ()= ( q 0 x 1 q 1 ),i.e.thelabeloftheedgefrom q 0 to q 1 ;If ( q 0 x 1 q 1 x 2 :::q p 1 )isgiven,then ( q 0 x 1 q 1 x 2 :::q p 1 x p q p )= ( q 0 x 1 q 1 x 2 :::q p 1 ) x p q + p wherefor x p = x p q + p denoteshorizontal k -concatenationof ( q 0 x 1 q 1 x 2 :::q p 1 )with q p := f q p ( i;j ): i = k 1 ; 0 j k 1 g ; andsimilarlywhen x p = x p q + p denotesthevertical k -concatenationof ( q 0 x 1 q 1 x 2 :::q p 1 )with b q p := f q p ( i;j ):0 i k 1 ;j = k 1 g : Theinputdatafor M F ( X ) isintheformofrectangulararraysofdata-inotherwords, blocks.Let k beapositiveintegerandsuppose B m;n isgivenfor m;n k .A k -phrase P B m;n issaidtobe acceptedby M F ( X ) ifandonlyifthereisapathin M F ( X ) such that ()= P .(Notethatifissuchthat ()= P ,then j j = m + n 2 k .)Set = f B m;n ( i;j ):0 i k 1 ; 0 j k 1 g ; (2.1.5) i.e.thelower-left k k subblockof B m;n ; andset = f B m;n ( i;j ): n k i n 1 ;m k j m 1 g ; (2.1.6) i.e.theupper-right k k subblockof B m;n Denition2.1.1 Block B m;n is acceptedbyautomaton M F ( X ) = f Q;E;s;t; g ifandonly if M F ( X ) acceptsall k -phrasesof B m;n thatstartwithstate q = andterminateinstate q = afterasequenceof k -concatenations = x 1 ;x 2 ; ;x p 2f *; g satisfying j j = n k and j j = m k .(For B m;n with m
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Proposition2.1.3 Let X beatwo-dimensionalshiftofnitetypehavingtheproperty F ( X )= A ( X ) .Lettheautomaton M F ( X ) beasconstructedabove.Then F ( X )= L ( M F ( X ) ) Proof. Toshowthat L ( M F ( X ) )= F ( X ),rstsuppose B m;n 2 F ( X )isgiven.If m
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(b) (a) 01 1 0 Figure2.3:Fullshiftasvertexshift(a)andedgeshift(b) singlepairofstatesifoneedgeisahorizontaltransitiona ndtheotheredgeisavertical transition.Notethat M F ( X ) alsofulllstherequirementsofanedgeshift. Wenowappealtodotsystemsinordertogeneratesomeexample softwo-dimensional nitestateautomataofreasonablesize.Wewillbeinterest edindotsystemsdenedby thegroup G = Z = 2 Z .Forsuchdotsystems,thesizeofthegraphrepresentingthe shift spaceisgivenbyCorollary1.3.3.Example2.1.4 Considerthedotsystem X knownintheliteratureastheThree-dot System,whichisdenedviatheshape S = f (0 ; 0) ; (0 ; 1) ; (1 ; 0) g .(Refertodiagram(a)in Figure1.3.)Forthisshiftspace X ,theset Q = F 2 ; 2 ( X )consistsof8statesasprovidedfor byCorollary1.3.3.Weusesolidlinestorepresenthorizont altransitionsanddashedlines torepresentverticaltransitions.Thedirectedgraphrepr esenting F ( X )(andtherefore X ) isdepictedinFigure2.4. 1 1 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0 Figure2.4:GraphrepresentingtheThree-dotSystem Asdiscussedinrelationtovertexandedgeshiftsintheonedimensionalcase,the labelsofbi-innitepathsofthegrapharepreciselythepoi ntsoftherepresentedshift 24

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space.Inonedimension,itisoftenofinterestwhetherdie rentbi-innitepathsmay beapresentationofthesamepointintheshiftspace;inothe rwords,whetherdierent bi-innitepathsmayhavethesamelabel.Ananalogousrepre sentationofbi-innitepaths intwodimensionswillrequirethefollowingdenition.Denition2.1.5 A grid-innitepath in M F ( X ) =( Q;E;s;t; )isdenedbyapairof maps, h : Z 2 Q and v : Z 2 Q ,andisdenotedasacollectionofstates f q [ i;j ] g (squarebracketsareusedtoavoidtheconfusionofthesymbo l q ( i;j ) = a 2 withthe stateassociatedtotheorderedpair( i;j ) 2 Z 2 )andthetransitionsthataccompanythese states. ... ... ... q [ 1 ; 1] *q [0 ; 1] *q [1 ; 1] q [ 1 ; 0] *q [0 ; 0] *q [1 ; 0] q [ 1 ; 1] *q [0 ; 1] *q [1 ; 1] ... ... ... Thiscollectionofstatesandtransitionsissuchthat 8 ( i;j ) 2 Z 2 ,thefollowinghold: i) q [ i;j ] 2 Q ii) q [ i;j ] *q [ i +1 ;j ] = e h isanedgein E h ,and iii) q [ i;j ] q [ i;j +1] = e v isanedgein E v ThediagraminDenition2.1.5commutesinthesensethatany blockthatisdescribed bysomeportionofagrid-innitepathisindependentoftheo rderinwhichtheedges aretraversed.Proposition2.1.6usesthisfacttoestablis haone-to-onecorrespondence betweenblocksinthelocallanguageandfactorsofgrid-in nitepaths. Proposition2.1.6 Giventhetwo-dimensionalshiftofnitetype X ,let M F ( X ) =( Q;E;s;t; ) beitsgraphrepresentation.Foragrid-innitepath of M f ( X ) ,let 0 besome 2 2 factor of comprisedoffouradjacentstates( q [ i;j ] ;q [ i +1 ;j ] ;q [ i;j +1] ;q [ i +1 ;j +1] )andthetransitions thatconnectthem.Then 0 representsaunique ( k +1) ( k +1) blockthatisrecognized by M F ( X ) 25

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Proof. ByDenition2.1.5,conditioni), q [ i;j ] 2 Q andisthereforea k k block.Denote by e h 1 ;e h 2 ;e v 1 ; and e v 2 theedges q [ i;j ] *q [ i +1 ;j ] ;q [ i;j +1] *q [ i +1 ;j +1] ;q [ i;j ] q [ i;j +1] ; and q [ i +1 ;j ] q [ i +1 ;j +1] ,respectively,whoselabelsareprovidedbyDenition2.1. 5,conditionsii) andiii).Thatis, 0 mayberepresentedbythefollowingdiagram. e h 2 q [ i;j +1] *q [ i +1 ;j +1] e v 1 e v 2 q [ i;j ] *q [ i +1 ;j ] e h 1 Thenormalized k -phrase P 1 describedby ( q [ i;j ] *q [ i +1 ;j ] q [ i +1 ;j +1] )has s ( P 1 )= q [ i;j ] and t ( P 1 )= q [ i +1 ;j +1] andisalmostcompletelydescribedbythesetwoblockswitht heexception ofthesymbolassignedto P 1 ( k; 0)by q [ i +1 ;j ] .Incomparison,thenormalized k -phrase P 2 describedby ( q [ i;j ] q [ i;j +1] *q [ i +1 ;j +1] )alsohas s ( P 2 )= q [ i;j ] and t ( P 2 )= q [ i +1 ;j +1] but hasasymbolmappedto P 2 (0 ;k )by q [ i;j +1] .(SeetheshadedlatticepointsinFigure2.5.) Therefore,the( k +1) ( k +1)blockrepresentedby 0 isindependentoftheorderof k -concatenation. Figure2.5:Comparisonof q*r s and q r 0 *s Corollary2.1.7 Let X beatwo-dimensionalshiftofnitetyperepresentedbythea utomaton M F ( X ) .Thenthereexistsaone-to-onecorrespondencebetweenpoi ntsin X and grid-innitepathsin M F ( X ) Proof. ApplyPropositions2.1.3and2.1.6. Inthesequel,weshallsometimesrefertoan m n factor 0 ofagrid-innitepathas a blockpath .Notethatan m n blockpathiscomprisedof mn statesanddescribesablock B 2 F ( X )ofsize( m + k 1) ( n + k 1).Thelabelingofablockpathshallbedenotedwith 26

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( 0 )= B andthestatesshallbedenotedby 0 ( i;j )for0 i n 1 ; 0 j m 1. SeeFigure2.6:Theshadingin(a)suggeststhatthe3 7blockofsymbolsmightbe representedbya2 6blockpathasindiagram(b)if M F ( X ) isdenedvia2 2states. Alternately,thesameblockofsymbolsmightbedescribedby a1 5blockpathif M F ( X ) weredenedvia3 3states. (a) (b) Figure2.6:Anexampleofablockpathwhere k =2 Example2.1.8illustratestheimportanceofthedenitiono ftheacceptedlanguage L ( M F ( X ) )beingbasedonblocksratherthanon k -phrasesalone.Inparticular,all k phrasesofanacceptedblockmustberepresentedbysomepath havingsource s ()= andtarget t ()= Example2.1.8 Considerthedotsystem X denedviatheshape S = f (0 ; 0) ; (1 ; 1) g and referredtointhispaperastheDiagonal-shiftSystem.(See shape(c)inFigure1.3.) Usingsolidlinestorepresenthorizontaltransitionsandd ashedlinestorepresentvertical transitions,adirectedgraphrepresenting X isdepictedinFigure2.7.Foreaseofreference, eachstate q 2 Q = F 2 ; 2 ( X )hasbeennumbered. Deneanormalized3 3block suchthat ( i;j )= 8<: 1for( i;j )=(1 ; 0) 0otherwise : Then = 2 F ( X ),where X istheDiagonal-shiftSystem.Byinspectionofthegraph M F ( X ) weseethatthe2-phraseof thatisdescribedbythelabelsofthepath q 1 q 0 *q 0 isacceptedby M F ( X ) ,althoughsuchadesignisnotasubsetofanypointfoundin X However,theblock itselfwillnotbeaccepted,asthereisnopathin M F ( X ) thatdescribes thesubblock 0 := f ( i;j ):0 i 2 ; 0 j 1 g andhence cannotbedescribedbya factorofanygrid-innitepathof M F ( X ) Sowehaveseenthatthe M F ( X ) constructionisatruevertex(edge)shiftinthesense thatpointsinthetwo-dimensionalshiftspace X arepreciselythelabelsofthegrid-innite 27

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0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 1 10 2 q 1 q 0 q 3 q 4 q 5 q 6 q 7 q Figure2.7:Graph M F ( X ) representingtheDiagonal-shiftSystem pathsfoundinthegraph.Therefore,Denition2.1.9canber ewordedasfollows. Denition2.1.9 Block B m;n is acceptedbytheniteautomaton M F ( X ) ifandonlyif thereexistsablockpath 0 in M F ( X ) having ( 0 )= B m;n 2.2RecognitionofSocShiftsby M F ( X ) Beforeexploringpropertiesofthegraphsrepresentingtwo -dimensionallanguages,wepoint outthepotentialusefulnessofthisnewtypeofniteautoma tonbyprovingthatthegraphs constructedhereaccuratelyrepresentcertaintwo-dimens ional strictlysocshifts ,i.e.soc shiftsthatarenotshiftsofnitetype.Specically,if X isatwo-dimensionalshiftofnite typeoverhavingthepropertythat A ( X )= F ( X ),thentheimageoftheblockcode : X Z 2 inducedbythe d d blockmap: B d;d ( X ) canberepresentedbythe underlyinggraphof M F ( X ) withstatesandlabelsadjustedaccordingly. Inone-dimensionallanguagetheory,agraphissaidtobe deterministic ifgivenalabel andavertex,thereisatmostonepathstartingatthegivenve rtexwiththespecied label.Notethatthegraph M F ( X ) isadeterministicgraphinthissense,sincegivenastate anda k ( k +1)label,atmostonehorizontaltransitionisspecied.(A nalogously,given astateanda( k +1) k label,atmostoneverticaltransitionisspecied.)Howeve r,when ablockcodeisappliedtothestatesof M F ( X ) ,thestatesneednolongerhavedistinct labelsandthegraphneednotbedeterministic.However,whi lethereneednolongerexist aone-to-onecorrespondencebetweengrid-innitepathsan dtheirlabels,itwillstillbethe 28

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casethatpointsintherepresentedtwo-dimensionalshifts pace Y arepreciselythelabels ofthegrid-innitepathsfoundinthegraph.Proposition2.2.1 Let X beatwo-dimensionalshiftofnitetyperepresentedby M F ( X ) = ( Q;E;s;t; ) ,andlet Y = ( X ) bethetwo-dimensionalshiftspacethatistheimageof X undertheblockcode inducedbytheblockmap .If M F ( X ) istheniteautomaton havingunderlyinggraph M F ( X ) withstateset Q 0 relabeledaccordingto Q 0 = ( Q ) andedge set E 0 relabeledaccordingto ( ( e )) ,then L ( M F ( X ) )= F ( Y ) Proof. Suppose B 0 2 F ( Y )isgiven.Since F ( Y )= ( F ( X ))= ( L ( M F ( X ) ),thereexists some 2 F ( X )suchthat ( )= B 0 .Soforthegraph M F ( X ) 2 L ( M F ( X ) )= F ( X ). However, M F ( X ) and M F ( X ) havethesameunderlyingedgesetsothat 2 L ( M F ( X ) ) ) B 0 2 L ( M F ( X ) ).Therefore F ( Y ) L ( M F ( X ) ). Forthereverseinclusion,say B 0 2 L ( M F ( X ) )isgiven.Inthiscase,theremustexist someblockpath 0 in M F ( X ) suchthat ( 0 )= B 0 .Usingtheunderlyinggraphof 0 ,we canndablockpath 0 withlabelsfrom M F ( X ) suchthat ( 0 )= 2 L ( M F ( X ) )= F ( X ). Inotherwords, ( )= B 0 2 F ( Y ).Therefore L ( M F ( X ) ) F ( Y ). IntheproofofProposition2.2.1,althoughtheblockpaths 0 and 0 areofthesame size,theblocks and B 0 thattheyrepresentneednotbe.Thisisduetothefactthat undertheblockcode ,theimage ( q )= q 0 ofeachstate q 2 Q willhavethickness k 0 =1+ k d .(Wemayalwaysassumewithoutlossofgeneralitythat k d since whenever F k;k ( X )denes X ,then F K;K ( X )alsodenes X forall K k .)Wenowrevisit Example2.1thatwasintroducedinthechapter-opener.That examplewasderivedfrom thefollowingexample,whichwasintroducedin[5]asanillu strationofthedicultyin presentingtwo-dimensionalsocshiftswiththetypeofgra phsknownatthetime.Inthis example,a2 2invertibleblockcodeisappliedtoashiftofnitetype X denedthrough aset F 2 ; 2 ( X )inordertocreateaconjugateshiftofnitetype X 0 denedthroughaset F 1 ; 1 ( X 0 );thereafter,a(non-invertible)1 1blockcodeisappliedto X 0 inordertocreate asocshiftspace.Example2.2.2 Dene X f a;b g Z 2 tobeatwo-dimensionalshiftofnitetypesuchthat foreverypoint x 2 X ,anyappearanceof b issurroundedby a 's.Thissubshifthasthe 29

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propertythat A ( X )= F ( X ):givenanyblock B 2 A ( X ),onecansimplysurround B with acongurationoftheplanepopulatedentirelywith a 's.Thelanguageoftheshiftspace X isdenedthroughthesetofallowed2 2blocks Q = F 2 ; 2 ( X )depictedinFigure2.8.By a aa a a ba a a ab a b aa a a aa b 1 3 4 p : = q : = 2 q : = q : = q : = Figure2.8:SetofveallowedblocksforsubshiftofExample 2.2.2 applyinga2 2invertibleblockcode, X isseentobeconjugateto X 0 f p;q 1 ;q 2 ;q 3 ;q 4 g Z 2 Considernextthesocsystem Y obtainedastheimageof X 0 underthe1 1blockcode denedby ( p )= p and ( q i )= q for i =1 ;:::; 4 : Byinspection,anyoccurrenceof q 1 in apoint x 2 X mustbeprecededhorizontallyby q 2 andfollowedverticallyby q 3 ,whereas q 3 mustbeprecededhorizontallyby q 4 .Thatis,inpoints y 2 Y ,the q symbolalways appearsin2 2blockscomprisedof q 's.Onecanpicturetheshiftspace Y tobethe collectionofallpointsthatresultfromconcatenationsof 1 1blockslabeled p and2 2 blockslabeledentirelywith q 's. Thesocsystem Y describedinExample2.2.2isstrictlysoc.Toseethis,sup pose towardsacontradictionthat Y isashiftofnitetype.Thentherewouldexistsome N 1 suchthatforthenormalized N N squareshape ,asucientconditionforapointto beintheshiftspace Y wouldbethatalltranslatesof hadadesignbelongingtotheset Q = F N;N ( Y ).Thatis, Y := f y 2f p;q g Z 2 : 8 v 2 Z 2 ; v ( y ) 2 Q g : (2.2.7) Nowconsideracongurationoftheplanepopulatedentirely by p 'swiththeexception ofthenormalized(2 N +1) 2rectangularshape T thatispopulatedwith q 's.More specically,considerthepoint y ( i;j )= 8<: q for0 i 1 ; 0 j 2 N p otherwise : (2.2.8) Then y issuchthat 8 v 2 Z 2 ; v ( y ) 2 Q ,whichwouldbesucientfor y 2 Y ;however, thiswouldcontradictthedenitionofthesiftspace Y sincenoconcatenationof2 2 blocksof q 'scouldproduceanoddnumberofrows(2 N +1)populatedwith q 's.So Y can 30

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notbeashiftofnitetypeandthereforemustbeastrictlyso cshiftspace. Therewasnosatisfactorygraphrepresentationof Y atthetime[5]waswritten.For exampleifwelet N =1in2.2.8,thentheuseofindependenthorizontalandverti cal scanningadmitsthepoint y 2 Y ,whichiserroneous.However,ifweusethe M F ( X ) constructiontorepresenttheshiftofnitetype X denedinExample2.2.2,thenthesoc system Y thatistheimageofthe1 1blockcode canbeaccuratelyrepresentedbythe underlyinggraphofFigure2.9. a a a a a b a a a a b a b a a a a a a b p qq q q 12 3 4 Figure2.9:Shiftspacewhereany b issurroundedentirelyby a 's Cautionmustbeobservedtoverifytheacceptanceofblocksi nthesocfactorspace Y basedonblockpathsratherthanmerelyasacollectionofpat hsacceptedby M F ( X ) Example2.2.3 Letthenormalized3 3block B begivenby B ( i;j )= 8<: p for( i;j ) 2f (0 ; 0) ; (1 ; 1) g q otherwise andlet Y betheshiftspaceofExample2.2.2.Informally, B = qqqqpq pqq Itisapparentthatblock B cannotbelongtothefactorlanguageof Y sinceitsdesign hasaperimetercomprisedofanoddnumberof q 'swhichcannotbeformedbythe concatenationof2 2blocksof q 's.Nowlet M F ( X ) betheunderlyinggraphofFigure2.9 withstates Q 0 = ( Q ).Thereadercanverifythatall1-phrasesofblock B areaccepted 31

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by M F ( X ) .However,thereisno3 3blockpath 0 of M F ( X ) havingthepropertythat ( 0 )= B .Toseethis,denotethestatesof M F ( X ) with p and q i for i =1 ; 2 ; 3 ; 4asin Figure2.9andlet 0 bea3 3blockpathof M F ( X ) thatisacandidatefor ( 0 )= B Forreference,usesubscriptstodistinguishthestatestha tcomprisetheblockpath 0 0 = q e q f q g q c pq d pq a q b Thenthefollowingconditionsspecifycertainstatesin 0 p q c *p ) q c = q 1 and q 1 q e ) q e = q 3 p*q a p ) q a = q 4 and q 4 *q b ) q b = q 3 Asestablishedthusfar,sixoftheninestatesin 0 aredictatedbythedesignof B 0 = q 3 q f q g q 1 pq d pq 4 q 3 Pathsinthegraphof M F ( X ) restrictthepossibilitiesfortheremainingthreestateso fthe blockpath 0 .Therearetwooptionsforthetoprowof 0 (i) q 3 *q 4 *q 3 or(ii) q 3 *q 2 *q 1 Forthefarrightcolumnof 0 therearealsotwooptions. (iii) q 3 q 1 q 3 or(iv) q 3 q 2 q 4 Bydenition,allpathscomprisingablockpathmustendinth esameterminalstate. Sobyinspection,onlypaths(i)and(iii)mightco-existwit hintheblockpath 0 .This processofeliminationrevealsthattheonlypossibilityfo rthestatesof 0 arethosegiven in(2.2.9).However,suchablockpathdoesnotexistin M F ( X ) sinceneither p q 4 nor p*q 1 isanedgein M F ( X ) 32

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0 = q 3 q 4 q 3 q 1 pq 1 pq 4 q 3 (2.2.9) Supposethat M F ( X ) representsatwo-dimensionalshiftspace X andlet A h and A v denotetheadjacencymatricesfor M hF ( X ) =( Q;E h ;s;t; )and M vF ( X ) =( Q;E v ;s;t; ), respectively.(Anadjacencymatrixofagraphisamatrixwhe rethe( i;j )entrydenotes thenumberofedgesinthegraphhavinginitialstate i andterminalstate j .)Intheliterature,therepresentationoftwo-dimensionalshiftspaces throughtheuseoftwoseparate graphs/matricesforhorizontalandverticaltransitionsi sdoneinasettingthatrequires A h tocommutewith A v [26].Thispropertyisnotnecessaryfortheunderlyinggrap hof ashiftspacerepresentedviathe M F ( X ) construction. Example2.2.4 Let X betheshiftspaceofExample2.2.2havingtheunderlyinggra phof Figure2.9.Withthestatesordered f p;q 1 ;q 2 ;q 3 ;q 4 g ,theadjacencymatricesfor M hF ( X ) = ( Q;E h ;s;t; )and M vF ( X ) =( Q;E v ;s;t; )donotcommute.Thatis, A h = 1010110101010001010100010 and A v = 1110000010000011110011100 inwhichcase A h A v = 2220122201000102220111100 6 = A v A h = 2120210101000102120221202 Forfuturereference,wecommentherethatthelabeledstate sinthegraphofFigure 2.9couldalternatelybedescribedasthesetof2 2blocksthatarepopulatedfromthe alphabet= f a;b g accordingtotherestrictionthateachblockcontain\atlea stthree 33

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a 's".Inasimilarfashion,othertwo-dimensionalshiftspac escanbeconstructedbyrst deningthesetofstatesaccordingtorestrictionsplacedo nasetoflocalizedpicturesand thendescribingthetransitionsthatexistbetweenthesest ates.Forexample,inChapter 4weshallmakereferencetotheshiftspacerepresentedbyag raphcomprisedof15states denedbyrestrictingthesetof2 2blockstothosehaving\atleastone a ".Inthe nextsection,weexaminetherelationshipbetweengraphsan dlocalizedpicturesfroma dierentangle:thatis,givenacollectionofverticesanda setofhorizontalandvertical edgesconnectingthem,whatlocalizedpicturesareeitherf orbiddenorforcedasaresult ofsucha(sub)graph? 2.3ForbiddenandForcedStructuresin M F ( X ) and M F ( X ) Inone-dimensionalalgebraicautomatatheory,itisknownt hatalanguage L isafactorial andprolongablerecognizable(FPR)languageiitisrecogn izedbyaniteautomaton M = f Q;E;s;t; g whereeverystate q 2 Q isbothinitialandterminalandbothsource s andtarget t aresurjectivefunctions[3].Takea(one-dimensional)ni teautomaton M =( Q;E;s;t; )andpartitionthetheedgesintohorizontalandverticaltr ansitions, denotedby M h =( Q;E h ;s;t; )and M v =( Q;E v ;s;t; )respectively.Wesaythata vertex q is a two-dimensionalsink if q isasinkwithrespecttoeither M h or M v a two-dimensionalsource if q isasourcewithrespecttoeither M h or M v ,and is blockisolated if q isisolatedwithrespecttoeither M h or M v Byconstruction,theniteautomata M F ( X ) and M F ( X ) representingtwo-dimensionalsoc shiftspaceswillcontainnoblock-isolatedvertices,notw o-dimensionalsinks,andnotwodimensionalsources.Likewise,anysubgraphrepresenting apointintheshiftspacemust containbothincomingandoutgoinghorizontalandvertical edgesforeachstate.However, sinceitisundecidablewhether A ( X )= F ( X )fortwo-dimensionalshiftspaces,sucient conditionsforagraphrepresentationtobethatofatwo-dim ensionalfactorialandprolongablerecognizable(2DFPR)languagemustalsobeundeci dable. 34

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Whenitisknownthat A ( X )= F ( X )foratwo-dimensionalshiftspace X ,the M F ( X ) constructionmakesitpossibletodocumentcertainsubgrap hsaseitherforbiddenorforced duetothenatureofincomingandoutgoingedgesthatconnect nearbystates.These observationsregarding(sub)graphsarebasedonthefactth atthegraph'stransitionsare denedforasetofstates Q havingdistinctlabels.Tobegin,Proposition2.3.1states thatcertaincombinationsofpathsoflength2willneverapp earinagraphunderlyingthe M F ( X ) construction. Proposition2.3.1 Let M F ( X ) =( Q;E;s;t; ) betherepresentationofthetwo-dimensional shiftofnitetype X .For q;q 0 ;q 00 ;r 2 Q ,supposethereexiststwo h -pathsin M F ( X ) having length 2 thatconnectstates q and r ,thatis, 1 = q*q 0 *r and 2 = q*q 00 *r Then 1 = 2 ;thatis, q 0 = q 00 Proof. Given q*q 0 and q*q 00 ,wehavethat q 0 (0 ;k 1) :::q 0 ( k 2 ;k 1) ... . ... q 0 (0 ; 0) :::q 0 ( k 2 ; 0) = q (1 ;k 1) :::q ( k 1 ;k 1) ... . ... q (1 ; 0) :::q ( k 1 ; 0) = q 00 (0 ;k 1) :::q 00 ( k 2 ;k 1) ... . ... q 00 (0 ; 0) :::q 00 ( k 2 ; 0) : Giventhat q 0 *r and q 00 *r ,wehavethat q 0 (1 ;k 1) :::q 0 ( k 1 ;k 1) ... . ... q 0 (1 ; 0) :::q 0 ( k 1 ; 0) = r (0 ;k 1) :::r ( k 2 ;k 1) ... . ... r (0 ; 0) :::r ( k 2 ; 0) = q 00 (1 ;k 1) :::q 00 ( k 1 ;k 1) ... . ... q 00 (1 ; 0) :::q 00 ( k 1 ; 0) : Sincethelabelsarethesame,itmustbethecasethat q 0 = q 00 (b) (c) (a) Figure2.10:In M hF ( X ) ,graphdiamondsoflength2areforbidden. Sowithrespecttothehorizontaltransitions E h of M F ( X ) graphdiamonds ofsize2 areforbidden.Figure2.10showssomeforbidden(sub)graph sfor M F ( X ) and M F ( X ) 35

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Proposition2.3.1cannaturallyberestatedintermsofvert icaltransitions.Furthermore,theresultholdstrueforanygraph M F ( X ) representingtheimageof X underablock code:althoughthelabelsonthestatesofthenewgraphneedn otbedistinct,theedges aredenedbasedonthetransitionsthatexistbetweenthedi stinctstatesinthepreimage andarenotalteredunderanyblockcodethatmaybeapplied.( Onlythelabelsonthe edgesarealtered.)Corollary2.3.2 For M F ( X ) and M F ( X ) ,all h -cyclesand v -cyclesoflength 2 mustbe disjoint. Takenseparately, M hF ( X ) and M vF ( X ) aretruevertexshifts,butwhenviewedasone graph,weseethattheedgesetof M F ( X ) doesnotprecludetheexistenceofboth q r and q*r .Thisresultsincertainlabelsbeingforceduponthestates basedonthedenitionoftransitionsthatoccurin M F ( X ) ,whichtherebyforcescertainlocalizedpictures. Proposition2.3.3anditscorollariesholdtrueforboth M F ( X ) and M F ( X ) .(Theproofis providedforthecase M F ( X ) only.) Proposition2.3.3 Supposethat ( Q;E;s;t; )= M F ( X ) ,wherethestates Q = F k;k ( X ) arepopulatedbychoicesfromthealphabet .If q;r 2 Q aresuchthat q r and q*r thenitmustbethecasethat q = BC AB and r = CD BC forsome A;B;C;D 2 ( k 1) ; ( k 1) Proof. Supposethestatesof M F ( X ) areofsize k k ;weshallemployahigherblockcode tochangethealphabetandtherebyrelabelstates q and r .Specically,renamesubblocks ofstate q asfollows. A := q (0 ;k 2) :::q ( k 2 ;k 2) ... . ... q (0 ; 0) :::q ( k 2 ; 0) B := q (1 ;k 2) :::q ( k 1 ;k 2) ... . ... q (1 ; 0) :::q ( k 1 ; 0) 36

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H := q (0 ;k 1) :::q ( k 2 ;k 1) ... . ... q (0 ; 1) :::q ( k 2 ; 1) C := q (1 ;k 1) :::q ( k 1 ;k 1) ... . ... q (1 ; 1) :::q ( k 1 ; 1) Inthesameway,renamesubblocksofstate r asfollows. E := r (0 ;k 2) :::r ( k 2 ;k 2) ... . ... r (0 ; 0) :::r ( k 2 ; 0) F := r (1 ;k 2) :::r ( k 1 ;k 2) ... . ... r (1 ; 0) :::r ( k 1 ; 0) G := r (0 ;k 1) :::r ( k 2 ;k 1) ... . ... r (0 ; 1) :::r ( k 2 ; 1) D := r (1 ;k 1) :::r ( k 1 ;k 1) ... . ... r (1 ; 1) :::r ( k 1 ; 1) Ineectthen, q and r maybeviewedas2 2statesunderthisnew(sliding)blockcode sothat q = HC AB and r = GDEF : Thecondition q*r forces E = B and C = G ,whiletheconditionthat q r forces C = G = F and H = E = B .Thetwoconditionstogetherimplythat q = BC AB and r = CD BC : AsdemonstratedintheproofofProposition2.3.3,wemayalw aysassumeahigher blockcodeasneeded.Wedosointhesequelwithoutanylossof generality.Inparticular, unlessstatedotherwise,allresultstofollowassume2 2statesareusedintheconstruction ofniteautomata. ThefollowingCorollarieswillbeusefulinthediscussiono fperiodicpoints. Corollary2.3.4 Supposethat ( Q;E;s;t; )= M F ( X ) .If q;r 2 Q aresuchthat q r and 37

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r*q ,thenitmustbethecasethat q = CA AB and r = AB BD : Corollary2.3.5 Supposethat ( Q;E;s;t; )= M F ( X ) .If q;r 2 Q aresuchthat q r;q* r ,and r q ,thenitmustbethecasethat q = BA AB and r = AB BA : r qs Figure2.11:HorizontalgraphtriangleofCorollary2.3.6 Corollary2.3.6 Supposethat ( Q;E;s;t; )= M F ( X ) .If q;r;s 2 Q aresuchthat s q;s*q;r q;q*r; and r*s ,then q = AAAA r = AAAB and s = AA BA Corollary2.3.6hasseveralvariations,allofwhichcontai n h -cyclesor v -cyclesoflength three.Werefertosuchsubgraphsas graphtriangles andshalltakeadvantageofthe appearanceofhorizontal(vertical)graphtrianglesinthe discussionofperiodicity:While notforbidden,graphtrianglesin M F ( X ) indicatemuchaboutthelabelsoncertainstates andtherebyforcesomelocalizedpicturesandforbidothers 38

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3Transitivity Nowthatwehavedenedagraphrepresentationforaclassoft wo-dimensionalshift spaces,wemayexaminetherelationshipbetweenadirectedg raphandthetwo-dimensional FPRlanguagethatitrepresents.Forexample,inone-dimens ionalsymbolicdynamics, itisknownthatanessentialgraph(oneinwhichallstatesha vebothincomingand outgoingedges)istransitiveifandonlyifitsedgeshiftis transitive[25].(Ingraphtheory, transitivegraphsarereferredtoasstronglyconnectedgra phs.)Aswehaveseen,there aredierenttypesoftwo-dimensionaltransitivityandmix ing,eachofwhichdenesan invariantpropertiesforconjugateshiftspaces.Inthisch apter,weemploydotsystems over Z = 2 Z toillustratethesevarioustypesoftwo-dimensionaltrans itivityandelaborateon resultsthatarepublishedin[16]regardingtransitivityi ntwo-dimensionallocallanguages denedbydotsystems.InSection3 : 1,itisshownthatthefactorlanguageofadot systemdenedbyarectangularshapelackingfreecellscann otbetransitiveincertain directionsalthoughthesamefactorlanguagewillbeunifor mlytransitive.Itisalsoshown thatthefactorlanguageofadotsystemdenedbyatriangula rshapewillbemixingbut notuniformlyso.InSection3 : 2,transitivityinthegraphof M F ( X ) isrelatedto(uniform) directionaltransitivityintherepresentedlanguage.Ama inresultisgiveninTheorem 3.2.6,whichstatesthatfortheshiftspace X having F ( X )= A ( X ),itisdecidablewhether F ( X )exhibitsuniformhorizontaltransitivityatagivendista nce K 3.1FactorLanguagesofDotSystemsDenedover Z = 2 Z Theinvestigationintothetransitivityofafactorlanguag eofadotsystemisconnectedto theexistence(orlackthereof)offreecellsinthedenings hape S .Thefollowingexample introducesthisidea.Example3.1.1 Let S = f (0 ; 0) ; (1 ; 0) ; (0 ; 1) ; (1 ; 1) g deneadotsystem X ,whichwe refertointhispaperastheFull-squareSystem.Dene 0 : S 7!f 0 g and 1 : S !f 0 ; 1 g 39

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with 1 ( i;j )=( i + j )mod2.Then f 0 ; 1 g F ( X ),buttheseblockscannotmeetin direction(1 ; 0).Toseethis,supposewewantedtoextendblock 0 inthe(1 ; 0)directionby horizontal2-concatenationwitha2 1block 0 toproducetheblock B 2 ; 3 2 F ( X ).Since ahorizontaltranslationoftheshape S intersectstheblock 0 atthecells f (1 ; 0) ; (1 ; 1) g and 0 (1 ; 0)+ 0 (1 ; 1)=0+0=0,theblock B 2 ; 3 willbeinthefactorlanguageonly if 0 (0 ; 0)= 0 (0 ; 1) 0 (0 ; 0)+ 0 (0 ; 1) (0mod2).Thisargumentcanberepeated indenitely,wherethesumofeachnewcolumnisrequiredtob eequivalenttozero.For 1 however, 1 (0 ; 0)+ 1 (0 ; 1)=0+1=1sothat 1 canneverappearasasubblockof anyblockextensionof 0 .Therefore,thefactorlanguage F ( X )cannotbehorizontally transitive.Thesameblockscanbeusedinasimilarargument todemonstratethat F ( X ) cannotbeverticallytransitive. Wendthatfordotsystems,theexistenceoffreecellsinthe (non-singleton)rectangle T thatminimallycontainstheshape S isanecessaryconditionforthefactorlanguage ofthedotsystemtobeverticallyand/orhorizontallytrans itive.Westatethehorizontal caseonly,astheverticalcasecanbeshownanalogously.Proposition3.1.2 Suppose X isadotsystemdenedthroughsome r c shape S such that c> 1 andlet T beanormalized r c shape.If T hasnofreecells,i.e. T = S ,then F ( X ) cannotbehorizontallytransitive. Proof. (TheWallpaperPattern)Summationsinthefollowingdiscus sionarecarriedout modulo2.Suppose X isadotsystemdenedthroughan r c rectangularshape S having c> 1.Notethatforall B 0 r;c 2 F ( X ),since P c 1 i =0 P r 1 j =0 B 0 r;c ( i;j )=0,wehavethat 8 i 0 2f 0 ; 1 ;:::;c 2 g ; i 0 X i =0 r 1 X j =0 ( B 0 r;c ( i;j ))= c 1 X i = i 0 +1 r 1 X j =0 ( B 0 r;c ( i;j )) : (3.1.1) Denotewith b i thesumofthebitsinthe i -thcolumn,i.e., b i = P r 1 j =0 B 0 r;c ( i;j )for i = 0 ;:::;c 1.Thentheequalitygivenin(3.1.1)impliesthat b 0 = P c 1 i =1 b i and P c 2 i =0 b i = b c 1 Nowutilizeblockextensiontoconstructanadditionalcolu mntotherightoftheoriginal block,therebyforming B r;c +1 2 F ( X ).Let b c bethesumofthebitsinthe c +1-stcolumn. 40

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Weseethatgivenany B 0 r;c 2 F ( X ), B 0 r;c B r;c +1 2 F ( X ) b 0 = c 1 X i =1 b i = b c (3.1.2) Thisisdepictedindiagram(a)inFigure3.1:Theshadedcolu mnsmusteachhavethe samesumasthatofthesumtakenoverthenon-shadedcolumnsc ombined.Ifweinspect thesequenceof0'sand1'sobtainedbysummingovereachcolu mnoftheoriginal r c block,wediscoverthatblockextensionswillinductivelyp roducethissamesequenceof columnarsumsinawallpaper-likepattern.Indiagram(b)of Figure3.1,columnswith thesameshadinghavethesamesum.Inparticular,if b i denotesthesumofthebitsin the i -thcolumnofablock B r;c + k for k> 0,then b c + k 1 = b q where q ( k modc). r,3c 0 b c B' r,c B r,c+1 b 0 b 0 b 0 b 1 b 1 b 1 (a) . . . . (b) B b Figure3.1:TheWallpaperPattern Finally,considerthespecicblock B 0 r;c 2 F ( X )thathas B 0 r;c (0 ; 0)= B 0 r;c ( c 1 ;r 1)=1 and B 0 r;c ( i;j )=0otherwise.Thenforanyblock B r;n 2 F ( X )ofwhich B 0 r;c isasubblock, B r;n cannotcontainthe r c blockconsistingofallzerosasasubblock.Yet,thissubblo ck ofallzerosiscontainedinthefactorlanguageofeverydots ystem.Thisdemonstratesthe existenceofapairofblocksthatcannotmeetindirection(1 ; 0)within F ( X ). Inthesequel,weshallmakefrequentreferencetowallpaper patterns.Bythis,we refertothefactthatfordotsystemsdenedbyan r c rectangularshapethatlacksfree cells,thesumstakenovertheindividualcolumnsofanyallo wedblockhavingheight r and length n>c mustobeyarepeatingpatternoflength c .Note,however,that c neednot betheminimumpatternlength,ascertainwallpaperpattern swillcontainsubpatterns. 41

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Example3.1.3 Suppose r =1intheconditionsofProposition3.1.2;forexample,say thedotsystem X isdenedbythe1 2shape S = f (0 ; 0) ; (1 ; 0) g .Thenitistrivialto showthat F ( X )exhibitsverticaltransitivity.However, F ( X )isnothorizontallytransitive since 0 : S 7!f 0 g and 1 : S 7!f 1 g cannevermeetindirection(1 ; 0).Rather,forany B m;n 2 F ( X ), B m;n ( i;j )= B m;n ( i +1 ;j )forall i 2f 0 ; 1 ;:::;n 2 g ;j 2f 0 ; 1 ;:::;m 1 g Inthiscase,thewallpaperpatternoflength c =2isalsoapatternoflength1. Theexistenceoffreecellsintherectangle T thatminimallycontainstheshape S ,while necessary,isnothoweverasucientconditionforthefacto rlanguageofthedotsystemto beverticallyand/orhorizontallytransitive.Toseethis, wewillgeneralizetherectangular shapesbyconsideringshapesthatappearasanysetofparall ellinescomprisedofthe samenumberofdots.Moreformally,giventwonon-zerovecto rs( u 0 ;v 0 )and( u 00 ;v 00 )with integercoordinateshavinggreatestcommondivisors d 0 = gcd ( u 0 ;v 0 )and d 00 = gcd ( u 00 ;v 00 ), wesaythatashape S isa parallelogramwithdeningnon-zerovectors ( u 0 ;v 0 ) and ( u 00 ;v 00 ) if S = f a d 0 ( u 0 ;v 0 )+ b d 00 ( u 00 ;v 00 ): a =0 ; 1 ;:::;d 0 ;b =0 ; 1 ;:::;d 00 g .Bythisdenition,a rectangularshapewithonecolumnoronerow(e.g.,Example3 .1.3)isnotaparallelogram, sinceoneofthedeningvectorswouldbethezero-vector.Si milarly,theshape(b)in Figure1.3isnottechnicallyaparallelogram.Noticethatm anyparallelogramsnaturally havefreecellsintherectangle T thatminimallycontainstheshape:forexample,the shapeinFigure1.3(c)isaparallelogramwithdeningvecto rs(1 ; 1)and( 1 ; 1)andhas freecellsinthe3 3rectangle T .TheshapeinFigure1.3(d)isaparallelogramdened withvectors(0 ; 2)and(2 ; 0)thatleavesnofreecellsinthe3 3rectangle T .Theproof ofProposition3.1.2caneasilybeadjustedtoshowthefollo wingCorollary. Corollary3.1.4 If S isaparallelogramwithdeningvectors ( u 0 ;v 0 ) and ( u 00 ;v 00 ) ,thenthe factorlanguage F ( X ) ofthedotsystemdenedby S isnottransitiveindirection ( u 0 ;v 0 ) norindirection ( u 00 ;v 00 ) Proof. Suppose S hasdimension r c ,let T bethe r c rectangularshapethatminimally contains S ,andlet 0 : T 7!f 0 g .Say i 0 =min f i :( i; 0) 2 S g ,andsay i 00 =max f i : ( i;r 1) 2 S g .Towardacounterexample,deneablock 1 2f 0 ; 1 g r;c \ F ( X )suchthat 1 ( i 0 ; 0)=1, 1 ( i 00 ;r 1)=1,and 1 ( i;j )=0otherwise.Then 0 ; 1 2 F ( X ),butitcan beprovensimilarlyasinProposition3.1.2that 0 cannotmeet 1 indirection( u 0 ;v 0 )nor 42

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direction( u 00 ;v 00 )within F ( X ). Soinparticular,theshape S = f (0 ; 0) ; (1 ; 0) ; (1 ; 1) ; (2 ; 1) g withdeningvectors(1 ; 0) and(1 ; 1),depictedinFigure1.3(e),leavesfreecellsintherecta ngle T ,yetthefactorlanguagedenedby S cannotbehorizontallytransitive.Thisexampleservestoi llustratethat theexistenceoffreecellsinthedeningshapedoesnotguar anteehorizontal(vertical)transitivity.Another(non-parallelogram)exampleisgivenby thefollowingdotsystem,which hasa\hole"inthedeningshape S = f (0 ; 0) ; (1 ; 0) ; (2 ; 0) ; (0 ; 1) ; (2 ; 1) ; (0 ; 3) ; (1 ; 3) ; (2 ; 3) g Let 1 2 F 3 ; 3 ( X )besuchthat 1 (1 ; 1)=1,and 1 =0otherwise;andlet 0 2 F 3 ; 3 ( X )be suchthat 0 ( i;j )=0forall0 i 2 ; 0 j 2.Usinganargumentsimilartothatof theWallpaperPattern,thereadercanverifythat 0 and 1 maynevermeetindirection (1 ; 0)within F ( X ). Thedenitionofdotsystemsdoesnotprecludeshapesthatar edisconnected.Infact, shapesthataredisconnectedmaydenedotsystemshavingfa ctorlanguageswiththesame propertiesasthosecorrespondingtoshapesthatare(simpl y)connected.Forexample,if ashape S isminimallycontainedinarectangle T thathasaroworacolumncomprised entirelyoffreecells,wesaythat S canbe reduced to S 0 where S 0 istheshapeobtained from S byerasingtherowsorcolumnsof T thatcontainonlyfreecells.Forexample, theshape canbereducedto andtheshape canbereducedto ThemethodofproofemployedintheproofofProposition3.1. 2alsorevealsthefollowing. Corollary3.1.5 Ifashape S canbereducedtoashape S 0 thatisarectangle,then S denesadotsystem X withfactorlanguagethatlackshorizontaland/orvertical transitivity. Proof. Suppose S hasdimension r c andwithoutlossofgeneralitysupposethat c> 1. (Weareinterestedinnon-singletonshapeswhere r> 1or c> 1.Notethatforadot 43

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system X denedthroughashapehaving r =1and c> 1, F ( X )willtriviallyexhibit verticaltransitivitybutnothorizontaltransitivity.)D ene 0 : T 7!f 0 g ,where T isthe r c rectangularshapethatminimallycontains S .Notethatif S canbereducedtoa shape S 0 thatisarectangle,then(0 ; 0) ; ( c 1 ;r 1) 2 S .Towardacounterexample,dene ablock 1 2f 0 ; 1 g r;c \ F ( X )suchthat 1 (0 ; 0)= 1 ( c 1 ;r 1)=1,and 1 ( i;j )=0 otherwise.Then 0 ; 1 2 F ( X ),but 0 cannotmeet 1 indirection(1 ; 0)within F ( X ). Toseethis,wecanmodifytheproofofProposition3.1.2toin cludeonlythosecellsthat intersecthorizontaltranslatesof S .Firstadjust(3.1.1)torerectthefactthatsumsare takenonlyoverthecellsofablock B r;c thatlieinhorizontaltranslatesof S 8 i 0 2f 0 ; 1 ;:::;c 2 g ; X w 2ff ( i 0 ( c 1) ; 0)+ S g\ T g ( B r;c ( w ))= X w 2ff ( i 0 +1 ; 0)+ S g\ T g ( B r;c ( w )) Inasimilarfashion,theremainderoftheproofofPropositi on3.1.2canbemodiedto rerectthatwhenusingblockextensiontoconstructadditio nalcolumnstotherightof ablock B r;c ,thesumistakenonlyoverthosebitsthatintersecthorizon taltranslates of S .Inthiscontext, 0 and 1 havewallpaperpatternsthatcannotmeetindirection (1 ; 0).Giventhat r> 1,thesameblockscanbeusedtoshowthat F ( X )lacksvertical transitivity. Corollary3.1.6 Ifashape S canbereducedtoashape S 0 thatisaparallelogram,then S denesadotsystemwithfactorlanguagethatlacksdirectio naltransitivityinthedirection ofthedeningvectors. Foroneclassofshapes,itistruethattheexistenceofafree cellisbothnecessary andsucientforthecorrespondingfactorlanguagetobebot hhorizontallyandvertically transitive.Theproofofthefollowingpropositionhighlig htstheframeworkofcarefully selectingboththepointoforiginandthedirectioninwhich tobegintheinductiveprocess of\llingablock".Proposition3.1.7 Fordotsystemsdenedthrougha 2 2 shape S ,theexistenceofa freecellissucientforthefactorlanguageofthedotsyste mtobebothverticallyandhorizontallytransitive.Thefactorlanguagecorrespondingt osuchshapesisneitheruniformly horizontallytransitivenoruniformlyverticallytransit ive. 44

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Proof. Wewillprovehorizontaltransitivitybyinductiononthenu mberofrows m ina pairofblocks.(Presentedwithtwoblocksfrom F ( X )havingdierentheights,wecan alwaysuseblockextensiontoextendtheshorterblocktoano therblockin F ( X )having greaterheight.)Theverticalcasecanbeshownanalogously Use m =2forthebasisstep,since f 0 ; 1 g 1 ;n F ( X ).Let 0 representthe2 2block consistingofallzeros,whichisanelementofthefactorlan guageforalldotsystems.To establishthebasisstep,wewillrstshowthatforeveryblo ck B 0 2 ;n 2f 0 ; 1 g 2 ;n \ F ( X ), B 0 2 ;n meets 0 indirection(1 ; 0)within F ( X ).Infact, 0 meetseveryblockindirection(1 ; 0)at distance 2.Say B 0 2 ;n : T 0 !f 0 ; 1 g :if P w 2ff ( n 1 ; 0)+ S g\ T 0 g ( B 0 2 ;n ( w ))=0,then B 0 2 ;n meets 0 indirection(1 ; 0)atdistance0within F ( X ),andtheresultistrivial.Soconsiderthe casewhen P w 2ff ( n 1 ; 0)+ S g\ T 0 g ( B 0 2 ;n ( w ))=1.Wecandeneablock B 2f 0 ; 1 g 2 ;n +3 \ F ( X ) thatminimallyencloses B 0 2 ;n and 0 inthefollowingway. B ( i;j )= 8>>><>>>: B 0 2 ;n ( i;j )for0 i n 1 ; 0 j 1 for i = n; 0 j 1 0 ( i ( n +1) ;j )for n +1 i n +2 ; 0 j 1 (3.1.3) In(3.1.3),*isdeterminedbythelocationofthefreecell(s )inthe2 2rectangle T :If neither(1 ; 0)nor(1 ; 1)isafreecell,let j 0 denotetheordinateofthefreecell(0 ;j 0 )andthen dene B ( n;j 0 )=1 ;B ( n; ( j 0 +1)mod2)=0;If(1 ; 0)isafreecell,then B 2 F ( X )only if B ( n; 1)=1,sowedene B ( n; 0)=0if(0 ; 1)isalsoafreecell,butdene B ( n; 0)=1 otherwise;Similarly,if(1 ; 1)isafreecell,then B 2 F ( X )onlyif B ( n; 0)=1,sowedene B ( n; 1)=0if(0 ; 0)isalsoafreecell,butdene B ( n; 1)=1otherwise.Byasymmetric argument,wecanalsoshowthat 0 meetsanyblock B 2 ;n indirection(1 ; 0)within F ( X ), andhenceanytwoblocksofheight2meetindirection(1 ; 0)within F ( X ). Next,assumebyinductionthatforanytwoblocks B 0 ;B 00 2 F ( X )havingheight m ,thereexistsablock B thatencloses B 0 and B 00 indirection(1 ; 0)within F ( X ).Then supposewearegiven B 0 m +1 ;u and B 00 m +1 ;v 2 F ( X ).Theproofiscompletedbytheinspection offourcasesbaseduponthelocationofthefreecell(s). casei):(0 ; 0)isafreecell.(Seediagram(a)inFigure3.2,wheredarkly -shadedregions indicatetheshape S .) 45

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b 1 b 2 (b) 1 b 2 b (a) Figure3.2:Theinductivestep Let 1 = f B 0 m +1 ;u ( i;j ):0 i u 1 ; 1 j m g ,andsimilarly,let 2 = f B 00 m +1 ;v ( i;j ):0 i v 1 ; 1 j m g .Werstwanttoextend 2 totheleftby onecolumnandtherebyconstructanewblock 0 2 .Notethat,basedontheshape S ,the assignmentof 0 2 (0 ; 0)isarbitrary:weassignthevalueonthiscellbaseduponen triesin theoriginalblock B 00 m +1 ;v andtheninductivelydenetheremainderofthecellsinthen ew columnupwardfromthiscell.Moreformally, 0 2 ( i;j ):= 8<: 2 ( i 1 ;j )for1 i v; 0 j m 1 B 00 m +1 ;v (0 ;j )+ B 00 m +1 ;v (0 ;j +1)for0 j m 1 : (3.1.4) (Usedenition(3.1.4)for 0 2 when(0 ; 0)istheonlyfreecell;If(1 ; 1)isalsoafreecell,do notincludetheaddend B 00 m +1 ;v (0 ;j +1)inthelastlineofthedenition.)Bytheinduction hypothesis,thereexists B m;n 2 F ( X )suchthat B m;n encloses 1 and 0 2 indirection(1 ; 0) within F ( X ).Nowlabelan( m +1) n rectangularshapeas T ,andbegintodene B : T !f 0 ; 1 g inthefollowingway. B ( i;j )= 8>>>>><>>>>>: B 0 m +1 ;u ( i;j )for0 i u 1 ; 0 j m B m;n ( i;j 1)for u i n 2 v; 1 j m 0 2 ( i (( n 1) v ) ;j 1)for i = n 1 v; 1 j m B 00 m +1 ;v ( i ( n v ) ;j )for n v i n 1 ; 0 j m (3.1.5) Tocompletetheproofoftherstcase,dene B ( i; 0)= B ( i; 1)+ B ( i 1 ; 1)for u i ( n 1) v .Thelocationofthefreecellguaranteesthattheassignmen toftheseremaining cellsdoesnotdisrupttheoverallacceptanceoftheblock B .Therefore,thereexists B thatencloses B 0 m +1 ;u and B 00 m +1 ;v indirection(1 ; 0)within F ( X ). Theotherthreecasesareanalogoustocase(i)whenslightmo dicationsaremade initially. 46

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caseii):(1 ; 0)isafreecell. Dene 1 and 2 asincase(i),butextend 1 onecolumntotherighttoconstructa newblock 0 1 beforeapplyingtheinductionhypothesis. caseiii):(0 ; 1)isafreecell.(Forexample,seediagram(b)inFigure3.2. ) Let 1 = f B 0 m +1 ;u ( i;j ):0 i u 1 ; 0 j m 1 g ,andsimilarly,let 2 = f B 00 m +1 ;v ( i;j ):0 i v 1 ; 0 j m 1 g .Thenextend 2 totheleftonecolumnto form 0 2 byrstdeningthecell 0 2 (0 ;m 1)= B 00 m +1 ;v (0 ;m 1)+ B 00 m +1 ;v (0 ;m ).(If(1 ; 0) isalsoafreecellasinFigure3.2(b),eitherdonotincludet headdend B 00 m +1 ;v (0 ;m 1) orusecase(ii)discussedabove.)Inductivelydenetherem ainingcellsinthenewcolumn downwardfromthiscellbeforeapplyingtheinductionhypot hesis. caseiv):(1 ; 1)isafreecell. Dene 1 and 2 asincase(iii),butinsteadextend 1 onecolumntotherightbefore applyingtheinductionhypothesis. Towardacounterexampletothequestionofuniformdirectio naltransitivity,consider theThree-dotSystem X andsupposethat F ( X )isuniformlyhorizontally(vertically) transitivewithindistance K .Let 0 bethe( K +2) ( K +2)blockofallzeros,and deneablock B K +2 ;K +2 2 F ( X )havingallentrieszerowiththeexceptionofthetop-right corner.Moreformally, B K +2 ;K +2 ( i;j )= 8<: 1if( i;j )=( K +1 ;K +1) 0otherwise (3.1.6) For i = f 1 ; 2 ;:::;K +1 g ,ablockextensionof i columnstotherightof B K +2 ;K +2 isuniquely determinedbytranslatesof S uptoheight K i forthe i thcolumn.(SeeFigure3.4for anexample.)Therefore, B K +2 ;K +2 canmeet 0 indirection(1 ; 0)(orindirection(0 ; 1)) within F ( X )onlyatsomedistance k K +1.Thesameblocksprovideacounterexample touniformverticalandhorizontaltransitivityinthefact orlanguagecorrespondingtothe shape S = f (1 ; 0) ; (0 ; 1) g .Allothershapesareanalogous. TheThree-dotSystemisamemberoftheclassofshapesencomp assedbyProposition 3.1.7.SincetheThree-dotSystemisknowntobemixing(see[ 19,20,30]),itisalready knowntobebothverticallyandhorizontallytransitive.Ho wever,notallshapesinthis 47

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classhavefactorlanguagethatismixing.Forexample,itis easilyseenthatthefactor languageofthedotsystemdenedbytheshaperepresentedin Figure1.3(b)isamember ofthisclassofshapesyetcannotbetransitiveindirection (1 ; 1).Itisinterestingtonote thatwhilethefactorlanguagescorrespondingtocertainsh apesmaylacktransitivityin specicdirections,theselanguagesmaystillbeuniformly transitiveingeneral. Proposition3.1.8 Let X beadotsystemdenedthroughan r c rectangularshape S (i.e. T = S ),andlet F ( X ) beitsfactorlanguage.Then F ( X ) isuniformlytransitive. F ( X ) ismixingifandonlyif S isasingleton. Proof. Werstconsiderthecasewhere r = c =1.If r = c =1,then S isasingleton and X containsonlytheonepointcomprisedofallzeros.Inthisca se, F ( X )istheset ofallrectangularblockscomprisedentirelyofzerosandas suchitis(uniformly)mixing. Conversely,rectangularshapescomprisedofmorethanoned otdenelanguagesthatlack horizontaland/orverticaltransitivity(Proposition3.1 .2)andhencecannotbemixing. Forallothercases,let B 0 ;B 00 2 F ( X )betworectangularblocksofsize m i n i ( i =1 ; 2), respectively.Toeasenotation,assumewithoutlossofgene ralitythat m i r and n i c Ifnot,rstuseblockextensionsasneededtoextend B 0 downand/ortotheleftandthen toextend B 00 upand/ortotherightbeforendingablock B thatenclosestheselarger blocks.Theblock B thatminimallyencloses B 0 and B 00 wouldthenbeasubblockof B Forthecasewhen r =1and c> 1,assumewithoutlossofgeneralitythat n 1 n 2 Bydenitionof F ( X ),theremustexistsomeblock 1 2 F ( X )thatisanextensionof B 0 by n 2 n 1 columnstotheright.Wedene B 2f 0 ; 1 g ( m 1 + m 2 ) ;n 2 \ F ( X )asfollows: B ( i;j )= 8<: 1 ( i;j )for0 i n 2 1 ; 0 j m 1 1 B 00 ( i;j m 1 )for0 i n 2 1 ;m 1 j m 2 + m 1 1 (3.1.7) Sincenotranslateof S willintersect 1 and B 00 simultaneously,weseethat B 2 F ( X ). Furthermore, B minimallyencloses B 0 and B 00 withdistanceatwhichtheymeetbeing0 (theblockstouch).Inotherwords, F ( X )isuniformlytransitive.Thecasewhen r> 1 and c =1issimilar. Fortheremainingcases,wewillshowthat B 0 and B 00 meeteachotheratdistance0 byenclosingtheminablock B ofsize( m 1 + m 2 ) ( n 1 + n 2 )whichwewillconstructby utilizingthesetofalltranslatesof S thatlieentirelywithintheblock B 48

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Step1:Begindeningtheblock B bydening B 0 asthebottom-leftcornerand B 00 asthetop-rightcorner.Theassignmentoftheremainingcel lsisaccomplishedbythe assignmentofacentrally-locatedtranslateof S :Consider S ,thetranslateof S whose top-leftcellhascoordinates( n 1 1 ;m 1 ).(SeeFigure3.3,wherethedottedrectangle denotestherectangularsizeofthetranslate S andthetop-leftcellisshaded.)Other 3 '' B B 4 2 1 Figure3.3:Theprocessofllinguptheblock B thanthiscell( n 1 1 ;m 1 ),allundenedcellsof B thatintersect S aretobepopulated withzeros.Thendene B ( n 1 1 ;m 1 )asneededtoobtaintheappropriatesumover S Step2:Nextwedene B inductivelyonthetop-leftportionoftheshapebyusingthe top-leftdotoftranslatesof S toguaranteetheoverallacceptanceoftheblock B .Consider thetranslate S +(0 ; 1).Withtheexceptionofthedotlocatedat( n 1 1 ;m 1 +1),alldotsof thistranslatearealreadyassigned.Therefore,dene B ( n 1 1 ;m 1 +1)asneededtoobtain theappropriatesumover S +(0 ; 1).Translatingupwardfor u =0 ; 1 ;:::;m 2 1,wemay sequentiallyllcellsbydening B ( n 1 1 ;m 1 + u )asneededforthetranslate S +(0 ;u ). Beginthenextsequenceat S +( 1 ; 0)andfor u =0 ; 1 ;:::;m 2 1,sequentiallydene B ( n 1 2 ;m 1 + u )asneededforthetranslate S +( 1 ;u ).Theremainingcellsinthe top-leftcornerofblock B aredenedsequentiallyinthesamemanner. Step3:Returnto S andnoticethateachofthecellsoriginallypopulatedwithz eros isthebottom-rightdotforsometranslateof S whosetop-leftdotliesintheregionofthe block B havingalreadybeendenedinanacceptablemanner.Wewilld ene B onthe remainingcellsinthebottom-rightportionoftheshapebyu singthebottom-rightdotof translatesof S inordertoguaranteetheoverallacceptanceofblock B .First,wedene thecellsthatcompletetherowstotherightofthosecellsth atwerepopulatedwithzeros instep1.Consider S 0 = S +(1 ;r 2).Thetranslate S 0 hasonlythedotlocatedatthe bottom-rightcornerundened,soatthiscellwedene B ( n 1 1+ c;m 1 1)asneeded 49

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fortheappropriatesumover S 0 .Translatingtotherightfor v =0 ; 1 ;:::;n 2 c ,we mayassign B ( n 1 1+ c + v;m 1 1)asneededfortheappropriatesumover S 0 +( v; 0). Continuingthisprocessfor w =1 ; 2 ;:::;r 2,wecandenerowsthatbeginwiththe translate S 0 +(0 ; w )andthenassign B ( n 1 1+ c + v;m 1 w 1)asneededtoobtain theappropriatesumover S 0 +(0 ; w )+( v; 0)for v =0 ; 1 ;:::;n 2 c Step4:Finally,wedenetheremainingcellsinthebottom-r ightportionoftheshape. Dene S 00 = S +( ( c 2) ; 1).Using w 0 =0 ; 1 ;:::;m 1 r ,wecandenerowsthat beginwiththetranslate S 00 +(0 ; w 0 )andthenassign B ( n 1 + v;m 1 r )asneededto obtaintheappropriatesumover S 00 +(0 ; w )+( v; 0)for v =0 ; 1 ;:::;n 2 1. Sotheentire( m 1 + m 2 ) ( n 1 + n 2 )block B 2 F ( X )canbelledinsuchawaythat B 0 isinthebottom-leftcornerand B 00 isatthetop-rightcorner.Notethat d ( B 0 ;B 00 )=0 regardlessofthesizeoftheblocks B 0 ;B 00 .Therefore, F ( X )isuniformlytransitive. SinceitisknownthatthefactorlanguageoftheThree-dotSy stemismixing,the inspectionofdotsystemsthatbehaveliketheThree-dotSys temisagoodplacetobegin furtherinvestigationsintofactorlanguageswhicharemix ing.Wesaythatan r c shape S is three-dotlike ifthedotsformasimply-connectedisoscelestriangle(suc hthat r = c forthe r c shape S ).Forexample, and areboththree-dotlike.Wecanapplythemethodofproofused inProposition3.1.8to showthefollowing.Proposition3.1.9 If S isashapethatisthree-dotlike,then S denesadotsystem X withfactorlanguagethatismixing.Proof. Withoutlossofgenerality,suppose S isan r r triangularshapeappearingin thebottom-leftcornerofthe r r rectangle T .Tobegin,let B m;n 2 F ( X )besuchthat m r;n r .Wenotesomepropertiesoftheseblocksfoundin F ( X ). 50

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determined by b * * * * * a a a a a a b bb * * * * ** # # ### # # # * * # ## b a predetermined arbitrary assignment defines additional row determined by a # Figure3.4:Extensionofablockintoatriangle i)Anyextensiontotherightof B m;n has(( m r +1)+( m r )+ ::: +1)dots predeterminedbytranslatesoftheshape,andanyextension upwardof B m;n has(( n r + 1)+( n r )+ ::: +1)dotspredeterminedbytranslatesoftheshape. ii)Followingthearbitraryassignmentof r 1cellsinacolumnabove B m;n ( n ( r 1) ;m 1), r 2cellsinacolumnabove B m;n ( n ( r 2) ;m 1) ;:::; and1cellabove B m;n ( n 1 ;m 1), B m;n maybeextendedalongthediagonalfromtheinsideouttotake onatriangulardesignthatoccursinsomepointoftheshifts pace X .(SeeFigure3.4 basedona4 4triangularshape S ,wherearrowsindicatethedirectioninwhichtoll.) iii)Additionalrowsbelowsuchatrianglemaybedenedbyth earbitraryassignment ofarowof r 1cellsdirectlybelowthedotsinthebottom-leftcorneroft hedesign(and/or additionalcolumnstotheleftofsuchatrianglemaysimilar lybedened). Nowconsideranytwoblocks B 0 m 1 ;n 1 ;B 00 m 2 ;n 2 2 F ( X ).Wewillshowthat B 0 meets B 00 ineverydirection( u;v )providedthat d ( B 0 ;B 00 ) >r +max f m 1 ;m 2 ;n 1 ;n 2 g .Beginby placingthebottom-leftcornerof B 0 at(0 ; 0)andthebottom-leftcornerof B 00 at( u;v ), andthenexpandbothblocksintothetriangulardesignsoutl inedin(i)and(ii)above. Theproofcanbecompletedbasedontheobservationthatrega rdlessoftheplacement, thereexistonlythreeshapesthatmightoccuratthepointof intersectionfortworight isoscelestriangles. Considerrstthecasewhere u;v 0andexpandthetriangularextensionof B 00 belowand/ortotheleftasoutlinedin(iii)untilthebottom -leftcornerofthistriangle touchesthetriangularextensionofblock B 0 .(Seediagram(a)inFigure3.5.)Theproofis 51

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completedbyllingintheblockthatminimallycontainsthe twotrianglesfromtheinside outinamethodsimilartothatusedinProposition3.1.8:wew illusethebottom-right dotintranslatesoftheshape S toassignvaluesunderneaththeuppertriangle,andwe willusetheupper-leftdotintranslatesoftheshape S toassignvaluestotheleftofthe uppertriangle.Specically,atthelocationwherethetwot rianglesmeet,arbitrarilydene enoughcellsintheleft-mostportionoftherowdirectlybel owtheuppertriangletobegin theinductionprocessthatwilldenetheremainderofthece llsinthisnewrow.(Note thatatthepointofintersectionthereisatmostonecelltha tisalreadydenedbythe lowertriangle,where r 1cellsmustbedenedinordertostartanewrow.)Inthisway, wemayinductivelydenecellsbytranslatingdownandstart ingoveronthefarleftof eachnewrow.Next,thecellstotheleftoftheuppertriangle willbedenedinductively beginningatthebaseoftheuppertriangleandworkingupeac hcolumnbeforetranslating lefttobeginatthebottomofthenextcolumn.Oncewehaveach ievedasingletriangular designthatenclosesthetwosmallertriangles,thecellsab ovethenewdiagonalmaybe lledinadiagonalmanner(seeshadedregionsinFigure3.5) :Beginatthetop-leftcorner withanarbitraryassignmentof r 1cellsalongthediagonalandthenworkdowneach diagonalbycheckingthebottom-rightdotintranslatesof S beforereturningtothetop tobeginthenextdiagonal. (a) (b) (c) Figure3.5:Possiblepositionsforintersectionofrightis oscelestriangles Inthecaseswhere u< 0or v< 0,extendingthetriangle(s)andinspectingthepoint wheretheymeetyieldsarectangularshapeandoneofthesame wedgeshapesencountered inquadrantone(seediagrams(b)and(c)inFigure3.5).Each oftheseshapesmay belledinamannerthatsatisesthegivendotsystemstruct urebybeginningatthe pointofintersectionofthetrianglesandworkingoutwardt ocreatealargertrianglethat containsthetwosmallertriangles.(Rectangularshapesat thepointofintersectionare tobescannedbycheckingthebottom-leftdotintranslateso f S .)Theremainingupper 52

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triangularportionoftheblockmaybelledalongthediagon alasinthequadrantone case. Intwo-dimensionalsymbolicdynamics,shiftspacesthatha veuniformlymixingfactor languagesareknowntohaveacertaindegreeofcomplexitywh ichismeasuredbythe topologicalentropyandisbasedonthenumberof n n blocksthatappearin F ( X ). Morespecically,the topologicalentropy ofthetwo-dimensionalshiftspace X is h ( X )=lim n !1 1 n 2 log 2 j F n;n ( X ) j : Proposition3.1.10 [6,27]If X isaninnitetwo-dimensionalshiftofnitetypewith factorlanguagethatisuniformlymixing,then X hasanon-zerotopologicalentropy. Example3.1.11 Let X betheThree-dotSystemandconsider F n;n ( X )forsome n 2. ByCorollary1.3.2, j F n; 1 ( X ) j =2 n .AsintheproofofProposition3.1.9,anyone-column designofheight n canbeextendedintoauniquetriangulardesignover n 1columns wherefor i 2f 1 ; 2 ;:::;n 1 g ,the i thcolumnhasheight n i .Againasintheproof ofProposition3.1.9,thechoiceofonesymbolatthetopofth e i thcolumndetermines thesymbolatthetopofeachconsecutivecolumn.(RefertoFi gure3.4foranexample.) Thereforeonlythetopsymbolofeachofthe n 1columnsmaybearbitrarilychosen whereby j F n;n ( X ) j =2 n +( n 1) .ThenthetopologicalentropyoftheThree-dotSystemis h ( X )=lim n !1 1 n 2 log 2 j F n;n ( X ) j =lim n !1 1 n 2 log 2 2 2 n 1 =lim n !1 2 n 1 n 2 =0 : SoalthoughtheThree-dotSystemhasfactorlanguagewhichi smixing,itcannotbe uniformlyso.Infact,ithasbeenshownthatalldotsystemsb elongtotheclassofshifts ofnitetypehavingtopologicalentropyequaltozero[20].Corollary3.1.12 If X isadotsystem,thenthefactorlanguage F ( X ) isnotuniformly mixing. 53

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3.2GraphRepresentations Similartotransitivityintheone-dimensionalcase,there isarelationshipbetweentypes oftransitivityfoundinatwo-dimensionallanguageandthe transitivityofthegraphthat recognizesthelanguage.Inthefollowing,recallthat E h denotesthesetofhorizontaledges withtransitionsdenotedby andthatanautomatonissaidtobetransitiveifitsgraph representationisstronglyconnected.Proposition3.2.1 Supposeatwo-dimensionalshiftofnitetype X issuchthat F ( X )= L ( M F ( X ) ) foranautomaton M F ( X ) =( Q;E;s;t; ) .If F ( X ) ishorizontallytransitive, then M hF ( X ) =( Q;E h ;s;t; ) istransitive.Inasimilarfashion,if F ( X ) isvertically transitive,then M vF ( X ) =( Q;E v ;s;t; ) istransitive. Proof. Weprovethehorizontalcaseonly.Let q 0 ;q 00 2 Q = F k;k ( X )beanytwostates.By denitionofhorizontaltransitivity,theremustexistsom eblock B k;n 2 F ( X )thatencloses q 0 and q 00 .Infact, B k;n canbeviewedasa k -phrase P having s ( P )= q 0 ;t ( P )= q 00 ,and length p = n 2.Thenbythedenitionof E h P canbeexpressedasasequence = q 0 *q 1 *:::q p 1 *q p ofverticesandhorizontaltransitions,wherefor1 i p q i 2 Q aresuchthat s ()= q 0 = q 0 t ()= q p = q 00 ,and q i 1 *q i 2 F ( X ).Inother words,isapathin M F ( X ) from q 0 to q 00 .Therefore, M hF ( X ) isatransitivegraph. Example3.2.2 ConsidertheFull-squareSystem X introducedinExample3 : 1 : 1,where X wasdenedvia S = f (0 ; 0),(0 ; 1),(1 ; 0),(1 ; 1) g .Thentheset Q = F 2 ; 2 ( X )consists of8states.Weusesolidlinestorepresenthorizontaltrans itionsanddashedlinesto representverticaltransitions.Inthegraphrepresenting F ( X ),twoseparatecomponents forthehorizontaltransitions(verticaltransitions)ill ustratethelackofhorizontal(vertical) transitivity.(SeeFigure3.6.) TheconversetoProposition3.2.1isfalse.Thatis,althoug htransitivityin M hF ( X ) is anecessaryconditionforhorizontaltransitivityofthelo callanguage F ( X ),itisnota sucientcondition.Toseethis,consideragraph M F ( X ) thatdisplaystransitivityinits horizontalcomponentandlet 0 = q 0 0 q 0 1 ::: q 0 p and 00 = q 00 0 q 00 1 ::: q 00 p beanytwo v -pathsin M vF ( X ) thathavethesamelength p .If F ( X )= L ( M F ( X ) )istobehorizontally 54

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1 1 0 1 00 10 1 0 1 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 11 1 Figure3.6:Full-squareSystem transitive,thenitisnotenoughthatforall i 2f 0 ; 1 ;:::p g thereexist h -paths i in M h F ( X ) having s ( i )= q 0 i and t ( i )= q 00 i ;rather,itmustalsobethecasethatthe h -paths inthecollection f i g overlapprogressivelytofromablockpath.Thisisillustra tedby thefollowingexample.Example3.2.3 Let S = f (0 ; 0) ; (0 ; 1) ; (0 ; 2) ; (1 ; 0) ; (1 ; 2) ; (2 ; 0) ; (2 ; 1) ; (2 ; 2) g deneadot system X .Weshallshowthatthelanguage F ( X )isnothorizontallytransitive,butthat thegraph M F ( X ) representing X istransitiveinbothitshorizontalandverticalcomponents, M hF ( X ) and M vF ( X ) ,respectively.ByCorollary1.3.3,thegraph M F ( X ) representing X wouldbecomprisedof2 8 states.Soratherthanconstructingagraphofsize256,wewi ll demonstratetransitivityinthegraph'shorizontalcompon entbyshowingthatany3 3 block B 3 ; 3 2 F 3 ; 3 ( X )= Q meetsthe3 3blockofallzeros 0 2 Q indirection(1 ; 0)within F ( X ).Thisisenoughtoshowthatthereisahorizontalpathconne ctinganytwostatesin thegraphsincetheverticalaxisofsymmetryfortheshape S allowsustouseasymmetric argumenttondapathfrom 0 toanarbitrarythirdblock B 0 3 ; 3 2 F 3 ; 3 ( X )= Q .(Bythe sametoken,thehorizontalaxisofsymmetryfortheshape S providesasimilarargument forshowingthat 0 meetseveryblock/state B 3 ; 3 2 F 3 ; 3 ( X )= Q indirection(0 ; 1)and viceversasothat M vF ( X ) istransitive.)Allsumsinthisexamplearecarriedoutmodu lo 2. Claim:Anyblock B 3 ; 3 2 F 3 ; 3 ( X )meets 0 indirection(1 ; 0)within F ( X ). ProofofClaim:InatechniquesimilartothatoftheWallpape rPatternsfoundinthe 55

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proofofProposition3.1.2,weinspectthesumofthebitsino ne-columnblockextensions ofahorizontal3-phrasethathassource B 3 ; 3 .Refertotheseone-columnextensionsas B i wherefor i 1each B i isa3 1blockwhosehorizontal3-concatenationwiththeexisting 3-phrase B 3 ; 3 *B 1 *B 2 *:::*B i 1 producesa3 (3+ i )blockin F ( X ).Say n 2istherstindexsuchthatthesumofthebits B n (0 ; 0)+ B n (0 ; 1)+ B n (0 ; 2)= b n isrequiredtobeoddinorderfortheblocktobeinthelanguag e F ( X ).(Ifthereis nosuch n ,thentheresultistrivialsincewecandene B 1 = B 2 tobecolumnsofall zerossothat B 3 ; 3 meets 0 indirection(1 ; 0)atdistance0.)Todesignanoddsumin column B n ,dene B n (0 ; 0)= B n (0 ; 1)= B n (0 ; 2)=1.Atthisdiscretetimestep, B n 1 intersectsthetranslate S +( n; 0)atthetwobits B n 1 (0 ; 0)and B n 1 (0 ; 2);forthenext translateof S ,thecolumn B n 1 willintersectthetranslate S +( n +1 ; 0)atthethreebits B n 1 (0 ; 0) ;B n 1 (0 ; 1)and B n 1 (0 ; 2).Therefore,theproofoftheclaimiscompletedby theinspectionoftwocasesforthetranslate S +( n +1 ; 0). 1 1 1 ? ? D 1 1 1 E E 1 1 0 E 1 1 1 D 1 1 0 D 0 0 0 E 1 1 1 1 1 0 E 0 0 0 E 0 0 0 E (a) (b) (c)(d) Figure3.7:One-columnextensionhavingevensum casei):Theone-columnextension B n +1 requiresanevensum. Inthiscase,designanevensumincolumn B n +1 bydening B n +1 (0 ; 0)= B n +1 (0 ; 1)=1 and B n +1 (0 ; 2)=0.Atthisdiscretetimestep, B n intersectsthetranslate S +( n +1 ; 0) atthesumofthetwobits B n (0 ; 0)+ B n (0 ; 2)=0,whereasforthenexttranslateof S thecolumn B n willintersectthetranslate S +( n +2 ; 0)atthesumofthethreebits B n (0 ; 0)+ B n (0 ; 1)+ B n (0 ; 2)= b n =1.Inthisway,boththe S +( n +2 ; 0)translate andthe S +( n +3 ; 0)translatewillrequireevensumssothatwedene B n +2 = B n +3 as columnsofallzeros,whichcompletestheproofoftheclaim. SeeFigure3.7. caseii):Theone-columnextension B n +1 requiresanoddsum. Inthiscase,designanoddsumincolumn B n +1 bydening B n +1 (0 ; 0)=1and B n +1 (0 ; 1)= B n +1 (0 ; 2)=0.Atthisdiscretetimestep, B n intersectsthetranslate S +( n +1 ; 0)atthesumofthetwobits B n (0 ; 0)+ B n (0 ; 2)=0,whereasforthenext translateof S ,thecolumn B n willintersectthetranslate S +( n +2 ; 0)atthesumofthe 56

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threebits B n (0 ; 0)+ B n (0 ; 1)+ B n (0 ; 2)= b n =1.Thetranslate S +( n +2 ; 0)willthen requireanevensum:dene B n +2 (0 ; 0)= B n +2 (0 ; 1)=1and B n +2 (0 ; 2)=0.Inthisway, boththe S +( n +3 ; 0)translateandthe S +( n +4 ; 0)translatewillrequireevensums sothatwedene B n +3 = B n +4 ascolumnsofallzeros,whichcompletestheproofofthe claim.SeeFigure3.8. 1 1 1 ? ? D 1 1 1 E D 1 0 0 D 1 1 1 D 1 0 0 D 1 1 0 E 1 1 1 1 0 0 D 1 1 0 D 0 0 0 E (a) (b) (c)(d) Figure3.8:One-columnextensionhavingoddsum Nowthatwehavedemonstratedthetransitivityofthehorizo ntal(andvertical)componentof M F ( X ) ,weshowthattheblock B 5 ; 3 2 F ( X )ofall1'scannotmeetthe5 3 blockofall0'swithin F ( X )indirection(1 ; 0)atanydistance. 1 1 1 1 11 1 1 11 1 1 1 111 0 0 00 00 0 00000 0 0 0 0 Figure3.9:A4 3blockof1'smeets4 3blockof0's Considerrstthe4 3blockofall1's:inorderforthisblocktomeetthe4 3block ofall0's,theremustexisttwooverlapping h -pathsin M F ( X ) ofthesamelengthwithboth h -pathshavingsource 1 (the3 3state/blockofall1's)andtarget 0 .Noticethat wheneveraone-columnblock B n havingasum B n (0 ; 0)+ B n (0 ; 1)+ B n (0 ; 2)= b n =1 ishorizontally3-concatenatedontoanexisting3-phrase, thesymmetryoftheshape S resultsinthecolumn B n intersectingthetranslate S +( n +2 ; 0)withthesamesum b n =1.Therefore,inordertohorizontally3-concatenatetwos ubsequentcolumns B n +1 and B n +2 havingcolumnarsums b n +1 = b n +2 =0,theone-columnblock B n +1 mustbe suchthat B n +1 (0 ; 1)=1whichrequiresthateither B n (0 ; 0)=1or B n (0 ; 2)=1butnot both.Soforoverlapping h -pathstoreachstate 0 simultaneously,theymustdescribea 4-phrasewithhorizontalrerectionsymmetry.(SeeFigure3 .9.)Forthe5 3blockofall 1's,suchrerectionsymmetrycannotexist.Therefore,bloc k B 5 ; 3 2 F ( X )ofall1'scan notmeetthe5 3blockofall0'swithin F ( X )indirection(1 ; 0)atanydistance. 57

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Theredoexistconditionsonthegraph M F ( X ) thataresucientforthefactorlanguage F ( X )tobehorizontallytransitive.Westatethehorizontalcas eonly,astheverticalcase isanalogous.Proposition3.2.4 Supposeatwo-dimensionalshiftofnitetype X issuchthat F ( X )= L ( M F ( X ) ) for M F ( X ) =( Q;E;s;t; ) .Ifthesubgraph M hF ( X ) istransitiveandifforevery fourstates q 0 ;q 1 ;q 0 0 ;q 0 1 having q 0 *q 1 ;q 0 q 0 0 ,and q 1 q 0 1 ,itnecessarilyfollowsthat q 0 0 *q 0 1 ,then F ( X ) ishorizontallytransitive. Proof. Let 1 ; 2 2 F ( X )beanytwoblocksinthelanguagedenedby Q = F k;k ( X ). Withoutlossofgenerality,wemayassumethatthetwoblocks havethesameheight.The proofbyinductiononthenumberofrows m willbesimilartotheproofofProposition 3.1.7.Let m = k forthebasisstep:then B 0 meets B 00 indirection(1 ; 0)within F ( X )since M hF ( X ) istransitive.Nowassumebytheinductionhypothesisthat 1 meets 2 indirection (1 ; 0)within F ( X )forallblocks 1 ; 2 2 F ( X ) \ m;n andconsider B 0 m +1 ;n 0 ;B 00 m +1 ;n 00 2 F ( X ).Let 1 = f B 0 m +1 ;n 0 ( i;j ):0 i n 0 1 ; 0 j m 1 g ,andsimilarly,let 2 = f B 00 m +1 ;n 00 ( i;j ):0 i n 00 1 ; 0 j m 1 g .Thenbytheinductionhypothesis, thereexists B m;n + n 00 2 F ( X )suchthat B m;n + n 00 encloses 1 and 2 indirection(1 ; 0) within F ( X ).Nowlabelan( m +1) ( n + n 00 )rectangularshapeas T ,andbegintodene B : T inthefollowingway. B ( i;j )= 8>>><>>>: B 0 m +1 ;n 0 ( i;j )for0 i n 0 1 ; 0 j m B m;n + n 00 ( i;j )for n 0 i ( n 1) ; 0 j m 1 B 00 m +1 ;n 00 ( i ( n ) ;j )for n i n + n 00 1 ; 0 j m (3.2.8) (Here,wemayassumethatasdenedthusfar,block B occursin X :asintheproof ofProposition3.1.7,beforedening B wemayrstextendtheblocks 1 and/or 2 by k 1columnstoensurethatthishappens.)Thecompletionofblo ck B isguaranteed bystatesfrom M F ( X ) ,beginningattheleft-mostundenedcellinthetoprowof B and workingacrossthetoprowbyfollowingedgesin E h .Moreformally,in M F ( X ) there existsstates q 0 = B f ( m 2 ;n 0 2) ; ( m 2 ;n 0 1) ; ( m 1 ;n 0 2) ; ( m 1 ;n 0 1) g q 0 0 = B f ( m 1 ;n 0 2) ; ( m 1 ;n 0 1) ; ( m;n 0 2) ; ( m;n 0 1) g ,and q 1 = B f ( m 2 ;n 0 1) ; ( m 2 ;n 0 ) ; ( m 1 ;n 0 1) ; ( m 1 ;n 0 ) g ;theexistenceofstate q 0 1 isguaranteedby A ( X )= F ( X ) 58

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andsuppliesthelabelfor B ( m;n 0 ),whilethecondition\foreveryfourstates q 0 ;q 1 ;q 0 0 ;q 0 1 having q 0 *q 1 ;q 0 q 0 0 ,and q 1 q 0 1 ,itnecessarilyfollowsthat q 0 0 *q 0 1 "guaranteesthe acceptanceoftheblockwiththisadditionallabel.Whenwer eachtheright-mostundened cellinthetoprowof B ,thestate q 0 1 = B f ( m 1 ;n 1) ; ( m 1 ;n ) ; ( m;n 1) ; ( m;n ) g isalreadypredetermined,whiletheacceptanceofalltrans itionsisguaranteedbythe conditioninvolvingthefourstatesof M F ( X ) TheconditionsofProposition3.2.4aresucientbutnotnec essaryforthelanguage F ( X )tobehorizontallytransitive.Forexample,thegraphofth eThree-dotSystem providedinFigure2.4doesnotmeettheconditionsofPropos ition3.2.4,althoughthe Three-dotSystemisknowntobemixingandisthereforebothh orizontallyandvertically transitive. Whilethegraphrepresentationsdiscussedhereemployhori zontalandverticaltransitions,thesegraphscanstillprovideinformationaboutt ransitivityinotherdirections. However,itwillbenecessarytoplaceboundsontheneighbor hoodinquestion. Proposition3.2.5 Supposeatwo-dimensionalshiftofnitetype X issuchthat F ( X )= L ( M F ( X ) ) foranautomaton M F ( X ) =( Q;E;s;t; ) .If F ( X ) isuniformlytransitivein direction ( u;v ) ,then M F ( X ) istransitive. Proof. Consideranytwostates q 0 ;q 00 2 Q = F k;k ( X ).As F ( X )isuniformlytransitive indirection( u;v ),theremustexistablock B 2 F ( X )thatminimallyencloses q 0 and q 00 indirection( u;v )suchthat d ( q 0 ;q 00 ) 0 ;v< 0.(If u =0,thentheresultistrivial,astherewouldexist somepathconnecting q 0 and q 00 in M vF ( X ) =( Q;E v ;s;t; ).)Weshalluseextensionto createablocktoolargetobecontainedinthe K -neighborhood.Considervector( K + k;h 0 ) 59

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q'' B (u,v) q' q'' q' k K(u,v) (K+k, h' ) q' (u,v) dB B'' h+k,k (a) (b) Figure3.10:Uniformtransitivityindirection( u;v ) thatisparallelto( u;v ).Then h 0 = v u ( K + k ).(Seeleftportionofdiagram(b)inFigure 3.10.)Considerablock B 00 h + k;k 2 F ( X )where h>h 0 byextending q 00 belowby h -many columnssothat q 00 appearsinthetopportionofthenewblock.Then q 0 meets B 00 within F ( X )indirection( u;v )atdistance d
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path.)Sosupposethat B 0 isrecognizedbysome( m 1) ( n 0 1)blockpath 0 and that B 00 isrecognizedbysome( m 1) ( n 00 1)blockpath 00 ,whereweassumewithout lossofgeneralitythat n 0 ;n 00 2.Horizontaltransitivitycanbeillustratedbyndinga blockpath in M F ( X ) suchthatthestatesintheinitial(left)columnof agreewith thestatesinthenal(right)columnof 0 andsuchthatthestatesinthenalcolumnof agreewiththethestatesintheinitialcolumnof 00 .Forexample,ifablockpathof length3overlapsthegivenblocksinthedesiredway,thenth edistanceatwhichthetwo blocksmeetwouldbezero,sincethisimpliesthattheorigin altwoblockstouch.Uniform horizontaltransitivityatdistance K canbesatisedbyndinga( m 1) ( k +3)block path forsome k K suchthat 8 j 2f 0 ; 1 ;:::;m 2 g 0 ( n 0 2 ;j )= (0 ;j )and ( k +2 ;j )= 00 (0 ;j ).Ablockpathoflength k +3isneededtorepresentablockof length k +4thatoverlapsthenaltwocolumnsofsymbols(comprising one v -pathin 0 ) andinitialtwocolumnsofsymbolsinblocks B 0 and B 00 ,respectively.(Seediagram(b)of Figure3.11,wheretheblockpath thatoverlapswith 0 and 00 isindicatedwithdashed lines,whilethedarkershadingindicatestheinitialandn al v -pathsof thatoverlapwith v -pathsof 0 and 00 .) B' B'' p p '' k symbols k+4 symbols (a) (b) Figure3.11:Blocksandblockpathsover2 2states Theorem3.2.6 If X isatwo-dimensionalshiftofnitetypehavingproperty A ( X )= F ( X ) ,thengivenadistance K ,thereisanalgorithmwhichdecideswhether F ( X ) has uniformhorizontaltransitivityatdistance K Theproofwillbecomprisedoffoursteps.Tobegin,theunifo rmityconditionisusedto placethequestionintotheframeworkofone-dimensionalla nguagessothatseveralwellknownresultsconcerningone-dimensionalrecognizablela nguagesmaybeapplied.We rstusethefactthatiftwoone-dimensionallanguagesarer ecognizable,thentheirunion 61

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isalsoknowntoberecognizable.Thesecondapplicationcon cernstheproduct L 1 L 2 of twoone-dimensionallanguages L 1 and L 2 givenby L 1 L 2 := f x 1 x 2 : x 1 2 L 1 ;x 2 2 L 2 g (Informally,astringofsymbolsbelongsto L 1 L 2 ifitcanbewrittenasastringin L 1 followedbyastringin L 2 .)Finally,wewillusetheresultthatitisdecidablewhethe rtwo one-dimensionallanguagesareequalornot.Proof. Step1:Let M F ( X ) =( Q;E;s;t; )beagraphrepresentationof F ( X ).For i 2f 1 ; 2 ;:::;k +2 g ,formtheset H i ofall h -pathsoflength i foundin M F ( X ) .Foreach H i ,formtheone-dimensionalniteautomaton M i inthefollowingway. Denethestatesof M i tobetheset H i Deneatransitionfromstate h = q 0 *q 1 *:::*q i 1 tostate h 0 = q 0 0 *q 0 1 :::*q 0 i 1 ifandonlyif 8 j 2f 0 ; 1 ;:::;i 1 g9 e v 2 E suchthat s ( e v )= q j and t ( e v )= q 0 j Anedgefrom h to h 0 isgiventhelabel h Notethateach M i isessentiallyaone-dimensionalvertexshiftrepresentin gthesetofall blockpathsoflength i +1thatrepresentblocksin F ( X ). Step2:Foreachniteautomaton M i ,formthelanguage L i bytakingproductsof thesequencesofstatesfoundinthelowerportionsofthers tandlastcolumnsofthe representedblockpathsinthefollowingway.Firstdisting uishbetweenstatesintheleft andrightcolumnsofablockbyformingtwodierentalphabet s Q and Q 0 .Thenlet L i := f j j g ,whereif = h 0 h 1 :::h m 2 L ( M i )isapathof\height" m ,thenforall j 2f 0 ; 1 ;:::;m 1 g j = (0 ; 0) (0 ; 1) ::: (0 ;j )= q 0 0 q 1 0 :::q j 0 isasequenceofstates foundinthebottompartoftherstcolumnof ;thatis, q s 0 isthesourceofthe h -path h s for s 2f 0 ; 1 ;:::j g .Inasimilarfashion, j =[ ( i 1 ; 0)] 0 [ ( i 1 ; 1)] 0 ::: [ ( i 1 ;j 0 )] 0 = q 0 0 i q 0 1 i :::q 0 j i isasequenceofstatesfoundinthebottompartofthelastcol umnofthesame blockpath ;thatis, q 0 s i isthetargetofthe h -path h s for s 2f 0 ; 1 ;:::j g .Nowdenethe recognizablelanguage L tobe L 1 [ L 2 [ ::: L k +2 .Thusdened, L representsthesetof allblocksthatcanmeetindirection(1 ; 0)atdistance k K in F ( X ).Asthelanguage L isformedfromthelabelsofbi-innitesequencesofthevert exshifts M i ,thelanguage L isnotnecessarilynite. 62

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Step3:Consider M vF ( X ) =( Q;E v ;s;t; ),therestrictionofthegraphrepresentationof F ( X )toverticaltransitionsonly.Relabel M vF ( X ) tobeavertexshiftwherethetransition q r islabeledwith q .(Weareinterestedincolumnarblockpathsratherthanthe symbolsfromthealphabet.)Denoteby L thelanguagerecognizedbythisrelabeled niteautomaton,andcreateasecondlanguage L 0 from L byattachingaprimetoeach state'ssymbol.Soifthealphabetfor L is Q = f q 1 ;q 2 ;:::;q f g ,thenthealphabetfor L 0 is Q 0 = f q 0 1 ;q 0 2 ;:::;q 0 f g .Nowdenetherecognizablelanguage LL 0 tobetheproductof L and L 0 .Thislanguage LL 0 representsthepossibilityofanytwoblocksin F ( X )meeting indirection(1 ; 0). Step4:Itisdecidablewhether LL 0 = L Theorem3.2.6canbeanalogouslyrewordedtodetermineif L ( M F ( X ) )= F ( X )exhibits uniformverticaltransitivityatdistance K .Transitivityinthehorizontaland/orvertical directionisofparticularinterest,sincethegraph M F ( X ) isdenedbasedonthesetypes oftransitions.Furthermore,thedenitionofperiodicity inthetwo-dimensionalcaseis basedonhorizontalandverticalmovement. 63

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4Periodicity Inone-dimensionalsymbolicdynamics,itisknownthatanys hiftspace X thatcanbe representedbyaniteautomatonmustcontainaperiodicpoi ntunderthe Z actionof coordinate-wisetranslation[25].Foratwo-dimensionals hiftspace X ,weareinterested inthe Z 2 actionoftranslatingelementsof X andtheperiodicpointsthatmayoccuras aresultofthisaction.Inthetwo-dimensionalcase,howeve r,evenashiftofnitetype canbeaperiodic(see[17]foranexampleofsuchashiftspace ).Inthischapter,welimit ourfocustotwo-dimensionalshiftspaceswherethefactorl anguage F ( X )isequivalent tothelocallanguage A ( X ).Forsuchtwo-dimensionalFPRlanguages,periodicpoints ofaspeciedperiodcanbeidentiedbythepresenceof h -cyclesand v -cyclesinthe graph M F ( X ) thatrepresents X .Inparticular,Theorem4.1.4statesthatif F ( X )= A ( X ) exhibitsuniformhorizontaltransitivityatdistance K ,then X mustcontainaperiodic pointwithaboundedleastperiod.Proposition4.1.5provid esanalgorithmforndingall periodicpointsuptoaspeciedboundintheshiftofnitety pe X representedbygraph M F ( X ) .Usingthisalgorithm,variousexamplesoftwo-dimensiona lperiodicpointsandthe (sub)graphsthatrepresentthemareprovidedinSection4 : 2. Beforediscussingperiodicpointsintwo-dimensionalshif tspaces,weneedamoreprecisenotionofwhatitmeansforapointtobeperiodicunderth eactionof Z 2 .Given thetwo-dimensionalshiftspace X x 2 X isperiodicofperiod( a;b ) 2 Z 2 nf (0 ; 0) g i x ( i;j ) = x ( i + a;j + b ) forevery( i;j ) 2 Z 2 .Furthermore,if x isperiodic,thenthereexists a;b> 0suchthat x ( i;j ) = x ( i + a;j ) = x ( i;j + b ) = x ( i + a;j + b ) forevery( i;j ) 2 Z 2 ,inwhichcase wesaythat x is doublyperiodic ofperiod( a;b ).If a and b aresuchthat a =min f a : x isdoublyperiodicofperiod( a;b ) g and b =min f b : x isdoublyperiodicofperiod( a;b ) g thenwesaythat( a;b )isthe leastdoubleperiod of x .Forexample,thepoint x inFigure 4.1isnotdoublyperiodicofperiod(3 ; 2)since x (0 ; 0) = A 6 = B = x (3 ; 0) ,although x is 64

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periodicofperiod(3 ; 2)since x ( i;j ) = A = x ( i +3 ;j +2) forevery( i;j ) 2 Z 2 asindicatedby theboxedsymbolsinFigure4.1. B B B A A A B B B A A A A A A B B B A A A B B B B B B A A A B B B A A A A A A B B B A A A B B B B B B A A A B B B A A A A A A B B B A A A B B B B B B A A A B B B A A A A A A B B B A A A B B B B B B A A A B B B A A A A A A B B B A A A B B B B B B A A A B B B A A A A A A B B B A A A B B B B B B A A A B B B A A A A A A B B B A A A B B B B B B A A A B B B A A A A A A B B B A A A B B B B B B A A A B B B A A A A A A B B B A A A B B B Figure4.1:Pointwithleastdoubleperiod(6 ; 4) 4.1DoublyPeriodicPointsinTwo-dimensionalShiftSpaces Proposition4.1.1guaranteestheexistenceofaperiodicpo intundercertainconditions.The propositionbeginswithidentifying h -cyclesofaxedlengthin M F ( X ) .(Alternatively,the propositioncouldidentify v -cyclesin M F ( X ) .) Proposition4.1.1 Let X beatwo-dimensionalshiftofnitetypehavingproperty A ( X )= F ( X ) andgraphrepresentation M F ( X ) =( Q;E;s;t; ) .Thereexistsapoint x 2 X being doublyperiodicofperiod ( a;b ) 2 Z 2 nf (0 ; 0) g ithereexist h -cycles f 1 ; 2 ;:::; b g in M F ( X ) (wherefor 1 b isdenotedby = q 1 *q 2 *:::*q a *q 1 )such that i)for 1 b j j = a ii)for 1 i b; 1 j a ,thereexists e v 2 E suchthat e v = q i j q ( i +1) j ,and iii)for 1 j a ,thereexists e v 2 E suchthat e v = q b j q 1 j Proof. Say x isdoublyperiodicofperiod( a;b ),andconsideradesignonthenormalized b a shape T whichwedenote x T = B b;a .Thenbythedenitionofdoublyperiodic, x T = x T +( a; 0) = x T +(0 ;b ) = B b;a .(See,forexample,Figure4.1).Inparticular,forthe normalized( b +1) a shape T 0 ,theblock B 0 b +1 ;a thatoccursin x mustbeacceptedby M F ( X ) insuchawaythatconditions(i),(ii),and(iii)arefullle d,sincethestatesof M F ( X ) havedistinctlabels. 65

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Fortheconverse,letcycles f 1 ; 2 ;:::; b g beasgivenandconsiderthefollowing repetitionofthesecycles: ... ... ... ... q 1 a *q 1 1 *q 1 a *q 1 1 q b a *q b 1 *q b a *q b 1 ... ... ... ... q 1 a *q 1 1 *q 1 a *q 1 1 q b a *q b 1 *q b a *q b 1 ... ... ... ... Thiscollectionofedgessatisestherequirementsforagri d-innitepath,andassuch, Corollary2.1.7guaranteestheexistenceofsomepoint x 2 X havingthepropertythat x ( i;j ) = x ( i + a;j ) = x ( i;j + b ) = x ( i + a;j + b ) forevery( i;j ) 2 Z 2 .Inotherwords, x isdoubly periodicofperiod( a;b ). InProposition4.1.1,neithercyclesnorstates(evenwithi nthesamecycle)needbe distinct.Considerapointhavingleastdoubleperiod( a;b ):Ifeachstateinsome b a blockpathrepresentingthepointweredistinct,thenthenu mberofstatesrequiredto representthepointwouldbe ab Corollary4.1.2 Giventhegraphrepresentation M F ( X ) ofatwo-dimensionalshiftofnite type X ,themaximumnumberofstatesrequiredtorepresentapointh avingleastdouble period ( a;b ) is ab Itispossibleforasubgraphoftwostatestobebothnecessar yandsucientforagraph M F ( X ) torepresentashiftspace Y thatcontainsaxedpoint.Forexample,considerthe subgraphof2statesconnectedinbothan h -cycleanda v -cycleasinCorollary2.3.5for 66

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somenon-deterministic M F ( X ) .If A = B sothatthepairofstateshavethesamelabels, thenthispairofstatessucetorepresentaxedpointofthe shiftspace.Whenviewed inthecontextof M F ( X ) ,however,thestatesmaybedistinctalthoughtheirlabelsa renot. (Sayoneofthestatesisdistinguishedbyaverticaltransit ionthattheotherstatelacks.) Ifthesocshiftspace Y representedby M F ( X ) hasnosinglestaterepresentingaxed point(thatis,nostatehasbothan h -loopanda v -loopassociatedwithit),thenthepair ofstatesrepresentasubgraphofminimumsizecapableofrep resentingaxedpointin Y ThissimpleexampleillustratesthatinProposition4.1.1, theconditionsforapoint x to bedoublyperiodicofperiod( a;b )-whilestillsucient-arenolongernecessaryinthe moregeneralsoccase.Thatis,themaximumnumberofstates requiredtorepresenta pointhavingleastdoubleperiod( a;b )mightexceed ab ifthelabelsonthestatesarenot distinct.Proposition4.1.3 Let Y beatwo-dimensionalshiftspacewithgraphrepresentation M F ( X ) =( Q;E;s;t; ) .Ifthereexistsapoint y 2 Y beingdoublyperiodicofperiod ( a;b ) 2 Z 2 nf (0 ; 0) g ,thenforsome k> 0 ,thereexist h -cycles f 1 ; 2 ;:::; kb g in M F ( X ) (wherefor 1 kb ,the h -cycle isdenotedby = q 1 *q 2 *:::*q ka *q 1 ) suchthat i)for 1 kb j j = ka ii)for 1 i kb; 1 j ka ,thereexists e v 2 E suchthat e v = q i j q ( i +1) j ,and iii)for 1 j ka ,thereexists e v 2 E suchthat e v = q kb j q 1 j Theexistenceofuniformhorizontaltransitivity(orunifo rmverticaltransitivity)for L ( M F ( X ) )= F ( X )guaranteesthattheshiftspace X hasaperiodicpoint. Theorem4.1.4 Let X beatwo-dimensionalshiftofnitetypehavingproperty F ( X )= A ( X ) .If F ( X ) exhibitsuniformhorizontaltransitivityatsomedistance K ,then X hasa periodicpointofleastdoubleperiod ( a;b ) forsome a K +2 Proof. Let M F ( X ) =( Q;E;s;t; )bethegraphrepresentationof F ( X )andconsidersome v -cyclecontainedin M vF ( X ) ,say = q 0 q 1 ::: q p = q 0 .(Sucha v -cyclemustexist since M vF ( X ) representsaone-dimensionalsocshiftspace,whichmustc ontainaperiodic point.)Denoteby B the( p +2) 2blockdescribedby ( ),andfor i 1considerthe set f B i g of( ip +2) 2blocksthatresultfromrepeatedlytravelingthe v -cycle .Since 67

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F ( X )exhibitsuniformhorizontaltransitivity, B i meets B i atadistance k K .Asthe set f B i g isinnite,theremustexist m K suchthatblocksfroma(countably)innite subset S f B i g allmeetthemselvesatdistance m .Foreach B j i 2 S ,let C 1 i ;C 2 i ;:::C j i ::: betheblocksthatconnect B j i with B j i .Since f C j i g isalsoinniteandsinceeachblock C j i haslength m ,theremustexistablock C J I withthepropertythatthe h -pathconnecting q 0 to q 0 whoselabeldescribesthebottomrowof C J I equalsthe h -pathconnecting q 0 to q 0 whoselabeldescribesthetoprowof C J I .(Thereexistonlyanitenumberof h -paths connecting q 0 to q 0 sincethegraph M F ( X ) isnite.)Finally,denetheblock B Ip +1 ;m +2 by B Ip +1 ;m +2 ( i;j )= 8<: B I ( i;j )for0 i 1 ; 0 j Ip C J I ( i 2 ;j )for2 i m +2 ; 0 j Ip (4.1.1) Then B Ip +1 ;m +2 containsallbutthetoprowofsymbolsinblocks B I and C J I sothat B Ip +1 ;m +2 ( i;j )= B Ip +1 ;m +2 ( i + a;j + b )with a = m +2 ;b = Ip +2isablockthatmakes upadoublyperiodicpointofperiod( a;b ). NoticethattheconversetoTheorem4.1.4isfalse.Forexamp le,onecaneasilynd periodicpointsintheFull-squareSystem,yetitwasshownt hatthefactorlanguageofthe Full-squareSystemlackshorizontal(andvertical)transi tivity.Whentheyexist,doubly periodicpointsofagivenperiod( a;b )canbelocatedforatwo-dimensionalshiftofnite type X throughuseofthe M F ( X ) graphrepresentationregardlessofwhether L ( M F ( X ) ) exhibitsuniformtransitivityornot.Proposition4.1.5 Given a;b> 0 andagraphrepresentation M F ( X ) =( Q;E;s;t; ) of thetwo-dimensionalshiftofnitetype X ,thereisanalgorithmwhichlocatesallpointsof X havingdoubleperiod ( a;b ) Proof. Byinspectionofthegraph M F ( X ) ,rstndall h -cyclesoflength a in M F ( X ) Distinguishcyclesbytheirinitialstate,makingpermutat ionsofthesamecycledistinct, sothattherewillbeatmost j Q j a cycles.Denotebythiscollectionofcycleshavinglength a ,andset j j = z .For1 z denote 2 by = q 1 *q 2 *:::*q a *q 1 Nextinspectpairsofcycles f ; g 2 .(Here f = g isnotprecluded,sothatthere existsatmost z 2 comparisons.)Ifitistruethat 8 j 2f 1 ; 2 ; ;a g ; 9 e v 2 E suchthat 68

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e v = q f j q g j ,thendenotethispairas r f g .Thecollectionofallsuchpairsistobedenoted by. Saythat r f g overlaps r f 0 g 0 i g = f 0 .Periodicpointscanbefoundbyprogressively overlappingpairsfoundin:Ifthereexists r f g 2 suchthat f = g ,thenthereexists apoint x 2 X beingdoublyperiodicofperiod( a; 1);ifthereexists r f g ;r f 0 g 0 2 such that r f g overlaps r f 0 g 0 insuchawaythat g 0 = f ,thenthereexistsapoint x 2 X being doublyperiodicofperiod( a; 2);andsoon. Givenagraphrepresentation M ofaone-dimensionalsocshift,itisfairlyeasyto locateperiodicpoints,asoneneedonlycheckthegraphforc ycles(whichmustexistin theone-dimensionalcase).Inthetwo-dimensionalcase,ev enashiftofnitetypemaybe aperiodic;whendoublyperiodicpointsdoexist,itisappar entfromProposition4.1.5and Proposition4.1.3thattheprocessoflocatingthesepoints willnotbesostraightforward,as onemustinspectthegraph M F ( X ) foroverlapping h -cyclesoflength ka .Morespecically, forasocsubshift Y andapoint y beingdoublyperiodicofperiod( a;b ),theremaynot existagrid-innitepathsuchthat( i;j )=( i + a;j + b )forall( i;j ) 2 Z 2 although y ( i;j )= y ( i + a;j + b )forall( i;j ) 2 Z 2 .Rather,itmaybethecasethatweneedto ndsome k> 1suchthat( i;j )=( i + ka;j + kb )forall( i;j ) 2 Z 2 ,asthestates neednolongerhavedistinctlabels.(If k =1,theconditionsofProposition4.1.3become thenecessaryconditionsofProposition4.1.1.)Forourdis cussionofperiodicpoints,we thereforefocusonthosedoublyperiodicpointsofperiod( a;b )thatarerepresentedbya grid-innitepathhaving( i;j )=( i + a;j + b )forall( i;j ) 2 Z 2 ,suchasthosefound in M F ( X ) representingashiftofnitetype X 4.2Examples Inthissection,weprovideexamplesofsubgraphsrepresent ingdoublyperiodicpoints ofleastperiod( a;b )giventhat a;b 2f 1 ; 2 ; 3 g .Inparticular,wecommentongraphs ofminimumsizecapableofrepresentingapointofleastdoub leperiod( a;b )giventhat a;b 2f 1 ; 2 ; 3 g ,andthenwegeneralizethesegraphsofminimumsizetoinclu decertain othertypesofdoublyperiodicpoints. Givenleastdoubleperiod( a;b ),Corollary4.1.2providesanupperboundonthenumber 69

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ofstatesrequiredtorepresentsuchapointinthegraphofat wo-dimensionalshiftofnite type.Furthermore,forall a;b 1thereexistssomegraphrepresentingapointforwhich thisupperbound ab isstrict,providedthealphabetislargeenoughforthenece ssary numberofdistinctstatestoexist.Thetaskofndingalower boundonthenumberof statesneededtorepresentapointofleastdoubleperiod( a;b )isabitmorearduous. (a)(b)(c)(d) Figure4.2:Vertexshiftsinone-dimensionalcase Intheone-dimensionalcase,supposeweconsiderthesetofa llbi-innitesequences (i.e.pointsinsomeone-dimensionalshiftspace)havingle astperiod a .Themaximum numberofstatesrequiredforavertexshifttorepresentsuc hapointwouldbe a .For example,let j j = a andlet X betheshiftspacerepresentedbyagraph M consistingof a statesconnectedinasimplecycle.(Figure4.2(d)issuchan examplefor a =3.)Atthe otherextreme,thevertexshiftofsize3havingtheunderlyi nggraphdepictedinFigure 4.2(c)isthesubgraphofminimumsizecapableofrepresenti ngsomepointofleastperiod a 3.(Graphs(a)and(b)ofFigure4.2arethesubgraphsofminim umsizecapableof representingpointsofleastperiod1and2,respectively.) Ifwelimitourselvestovertex shiftsrepresentingone-dimensionalsubshifts,theninte rmsofgraphshavingminimum size,thereareexactlyfournon-isomorphicsubgraphscapa bleofdepictingapointhaving leastperiod a 2f 1 ; 2 ; 3 g .ThesearedocumentedinFigure4.2. Accordingtothestandarddenitionofdoublyperiodic,ax edpointinatwo-dimensional shiftspacehasleastdoubleperiod(1 ; 1).Usingthe M F ( X ) constructionwith2 2states, xedpointsaretheonlydoublyperiodicpointsthatcanbere presentedbyasubgraph comprisedofasinglestateduetotheneedforbothan h -loopanda v -loopatthesame state.Fortwo-dimensionalshiftsofnitetypethen, ab =1isboththemaximumand minimumgraphsizeneededtorepresentapointofleastdoubl eperiod( a;b )=(1 ; 1). Sinceagraphofsize1canonlyrepresentaxedpoint, ab =2isboththemaximum andminimumgraphsizeneededtorepresentapointofleastdo ubleperiod(2 ; 1)or(1 ; 2) fortwo-dimensionalshiftsofnitetype.Inregardstoothe rgraphsofsize2,recallthat accordingtoProposition2.3.1,graph(c)ofFigure4.2isaf orbidden(sub)graphforany 70

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M F ( X ) having2 2statesthatrepresentsatwo-dimensionalshiftspace.The refore,doubly periodicpointsrepresentedbytwostatescanonlyhave h -cyclesoflength1or2-notboth, andlikewisefor v -cycles.Forexample,ifwecanlocateasubgraphof M F ( X ) comprisedof twodistinctstatesthathave v -loopsandthatareconnectedina h -cycleoflength2,then wewillhavelocatedapointintheshiftspacehavingdoublep eriod(2 ; 1).(Seegraph(a) ofFigure4.3.)Thesetwo-dimensionalpointsareratheruni nteresting,astheytakeonthe appearanceofaninnitenumberofcopiesofaone-dimension alpointofperiod2thatare \stacked"verticallyoneupontheother.Forexample, ... ... ... ... abab abab ... ... ... ... issuchapoint.Therefore,unlessotherwisenoted,whendis cussingpointshavingleast doubleperiod( a;b )weshallassumethat a;b> 1. (b) (a) Figure4.3:Pointsofdoubleperiod(2 ; 1)and(1 ; 2) Itispossible,however,fortwostatesinagraph M F ( X ) tobecapableofrepresenting atwo-dimensionalpointofdoubleperiod(2 ; 2).Thisgraphofminimumsizecapableof representingapointofleastdoubleperiod(2 ; 2)isnotthegraphofmaximumsizecapable ofrepresentingapointofleastdoubleperiod(2 ; 2). Proposition4.2.1 Usingthe M F ( X ) constructiontorepresentashiftofnitetype X thereexistexactlytwo(uptoisomorphism)subgraphscapab leofrepresentingapointhaving leastdoubleperiod (2 ; 2) Proof. Considergraph(b)ofFig4.4containingfourstates.ByProp osition4.1.1,any x 2 X havingleastdoubleperiod(2 ; 2)hassuchasetoffourstateswiththecorresponding transitionsimpliedbysome2 2factor 0 ofthegrid-innitepathrepresentingthepoint x .Denoteby G thisgraphcorrespondingtotheblockpath 0 .Tondasmallergraph 71

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representingapointofdoubleperiod(2 ; 2),wecouldidentifymultipleappearancesofthe samestatewithin G andthen\glue"thesestatestogethertocreateagraphwithd istinct statesasrequiredintheconstructionof M F ( X ) .Byinspection,however,wecanseethat combininganyonepairofstatesingraph(b)ofFigure4.4wou ld\collapse"acycleand leadtothecreationofaforbiddensubgraph.(Forexample,i fwegluestate1tostate4, nondisjoint h -cyclesoflength2areformed.)Wemay,however,gluemultip lepairsofstates evenifglueingthesestatesproducedforbiddensubgraphsw henappliedindependentlyof oneanother.Notice,though,thatifwegluestate1tostate2 andgluestate3tostate4, wecreategraph(b)ofFigure4.3.Thatis,thegraphnolonger representsapointhaving doubleperiod(2 ; 2).Soweseethattheonlywaytogluestateswhilemaintainin gcycles oflength2withoutcreatingaforbiddensubgraphistoglues tate1tostate4andglue state2tostate3.Theresultisgraph(a)ofFigure4.4.Anyfu rtherreductioninthesize ofthegraphwouldcollapsebothcyclesintoasinglestatere presentingaxedpoint. (a) (b) 1234 Figure4.4:Subgraphsrepresentingpointofdoubleperiod( 2 ; 2) Wewillcontinuetoemploytheconstructionintroducedinth eproofofProposition 4.2.1tocreateadeterministicgraphfromablockpath.That is,startingwithagraphof size ab representingthemaximumnumberofpossiblestatesasprovi dedbyCorollary4.1.2, weseekoutpossiblemultipleoccurrencesofthesamestate. Thegraphisthendecreased insizebyglueingtogethermultiplecopiesofthestateinto asinglestatethatpreserves alltransitions.Moreformally,if G = f Q;E;s;t; g issuchthat ( q i )= ( q j )forsome i 6 = j ,thenconstructadeterministicgraph G 0 = f Q 0 ;E 0 ;s;t; g from G inthefollowing way.Denethesetofstates Q 0 baseduponthesetofstates Q :if q i ;q j 2 Q aresuch that ( q i )= ( q j )(here, i = j isnotprecluded),thenthisdenesauniquestate q 0 2 Q 0 having ( q 0 )= ( q i )= ( q j ).Denetheedgeset E 0 basedupontheedgeset E :if e 2 E hassource s ( e )= q s andtarget t ( e )= q t ,thenthisdenesauniqueedge e 0 2 E 0 having 72

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s ( e 0 )= q 0 s andtarget t ( e 0 )= q 0 t ,where ( q 0 s )= ( q s )and ( q 0 t )= ( q t ).Inparticular,when ( q i )= ( q j )forsome i 6 = j ,theedgein G having s ( e )= q i and t ( e )= q j willappearasa loopin G 0 .Werefertothisprocessofconstructingasmaller,determi nisticgraphfroman existing(non-deterministic)graphas stateamalgamation .Thecasewhereanentirecycle isreducedtoasinglestateisreferredtoas collapsingacycle Atwo-dimensionalpointofperiod(2 ; 2)representedby4statesasdepictedingraph (b)ofFigure4.4canbelocatedinFigure2.9overthefour q i states( i =1 ; 2 ; 3 ; 4).A subgraphovertwostatesaspicturedingraph(a)ofFigure4. 4canbelocatedinthe Diagonal-shiftSystem.(SeeFigure2.7,states q 4 and q 5 .) Alternately,wecouldhaveappliedCorollary2.3.5tonegat etheexistenceofanydoubly periodicpointofperiod(2 ; 2)overasubgraphofsize3.Wedosohereforfuturereference Lemma4.2.2 Givenashiftofnitetype X ,no x 2 X ofleastdoubleperiod ( a; 2) can berepresentedbyasubgraphof M F ( X ) comprisedofexactly 3 states. Proof. ByProposition4.1.1,agraphrepresentingapointofdouble period( a; 2)would needtoexhibit v -cyclesoflength2,butCorollary2.3.2statesthat v -cyclesoflength2 mustbedisjoint.Thisimpliesthattheonlyvalidcombinati onof v -cyclesoflength2over 3stateswouldbeone v -loopandone v -cycleoflength2.Let q;r and s bethethreestates inthesubgraph,andsupposethatthe v -cycleoflength2connects q and r ,while s hasthe v -loop.Theproofiscompletedbythepresence-orlackthereo f-ofahorizontaltransition between q and r .Supposesuchatransitionexists.Inthiscase,thelabelso fthestates mustbethosegiveninCorollary2.3.5.Ifthiswereso,thent herewouldbenostatethat couldsatisfytheconditionfor s (hasa v -loop)whileatthesametimecoexistinginan h -cyclewith q and/or r .Ontheotherhand,ifnohorizontaltransitionexistsbetwe en q and r ,thenoneof q or r musthavean h -loopforanedge,whiletheothermustcomprise an h -cycleoflength2with s .(Otherwise,either h -cyclesoflength2areconnected,orthe threestatesformtwoseparatesubgraphs.)Withoutlossofg enerality,suppose q hasthe h -loop.Thenthe v -loopat s andthe v -cycleoflength2connecting q and r implythat s = CDCD q = AA BB and r = BB AA : Then s*r ) A = B = D and r*s ) A = B = C .Thatis, A = B = C = D sothat 73

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q = r = s ,andtheonlypointsofashiftspacethatcanberepresentedb yasinglestate arexedpoints. Naturally,Lemma4.2.2couldalsobestatedintermsofpoint shavingleastdouble period(2 ;b ).However,hereandinthesequel,weneedonlyfocusonpoint sofleast doubleperiod( a;b )where a b ,sinceallotherdiscussionsareanalogous. Whenmax f a;b g =3,onemusttakecarenottocreategraphdiamondsofsize2du ring theamalgamationprocess.Proposition4.2.3 Usingthe M F ( X ) constructiontorepresentashiftofnitetype X thereexistexactlythree(uptoisomorphism)subgraphscap ableofrepresentingapoint havingleastdoubleperiod (3 ; 2) Proof. Beginwiththecollectionof6statesthatwouldappearina2 3blockpath 0 of agrid-innitepathrepresentingsomedoublyperiodicpoin tofperiod(3 ; 2).Formgraph G fromtheblockpath 0 asbefore.Stateswithinthe h -cyclesoflength3mustallbe distinct-otherwisetherewouldexistnondisjoint v -cyclesoflength2.Avalidsubgraph iscreatedbycollapsingoneofthe v -cyclesoflength2,resultinginagraphconsistingof5 states.(SuchasubgraphcanbefoundinthegraphgiveninFig ure2.9,wheretheedgeset doesnotcontainthe h -loopatthestatelabeled p .)However,ifweattempttocollapsea second v -cycle,theresultinggraphcomprisedof4stateswouldcont ainahorizontalgraph diamondofsize2,whilecollapsingallthree v -cycleswouldyieldtheuninterestingpointof period(3 ; 1).Byreturningtotheoriginalgraph G comprisedof6states,anotheroptionis discoveredthroughtheamalgamationofcertainstatesinad jacent v -cycles.Forexample, asdepictedinFigure4.5,wecouldamalgamatestate1withst ate5(denotethisasthe 1 = 5state)andamalgamatestate2withstate4(denotedasthe2 = 4state). Finally,byLemma4.2.2,ithasbeenshownthatadoublyperio dicpointofperiod(3 ; 2) cannotexistoverthreestates,whichprohibitsanyfurther amalgamationpossibilities. Soincludingtheoriginalgraphofsize6,thereexistthreed istinctgraphscapableof representingdoublyperiodicpointsofperiod(3 ; 2). Inparticularthen,forapoint x havingleastdoubleperiod(3 ; 2),agraphpresentation 74

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(a) (b) 64 1235 146 3 25 (c) 3 1/5 6 2/4 Figure4.5:Stateamalgamation of x mustbecomprisedofatleastfourstates.TheFull-squareSy stemcontainsasubgraph ofthissizerepresentingapointintheshiftspacehavingle astdoubleperiod(3 ; 2). Example4.2.4 ConsidertheFull-squareSystem X denedviatheshape S = f (0 ; 0), (0 ; 1),(1 ; 0),(1 ; 1) g andgraphedinFigure3.6.Sayweweretodeneasubgraphbyr st bisectingthegivengraphwithaverticalline l drawnthroughthecenterofthegraphand thendiscardingtherightsideofthebisectedgraphaswella sthetwo h -loopsontheleft sideofthegraphandallverticaledgesintersecting l .Thentheremainingsubgraphwould representatwo-dimensionalpointofperiod(3 ; 2). IntheproofofProposition4.2.3,apairofamalgamationswe reappliedtonearbynonadjacentstateslocatedinadjacent v -cycles.AnapplicationofProposition2.3.3reveals othergeneralgraphstructuresthatappearwhenstatesofab lockpathareamalgamated withnearbystates.Proposition4.2.5 Let besomegrid-innitepathin M F ( X ) =( Q;E;s;t; ) representingthepoint x 2 X havingdoubleperiod ( a;b ) for a;b 2 ,andlet 0 beany 2 2 block pathof where 0 = q [ i;j +1] *q [ i +1 ;j +1] q [ i;j ] *q [ i +1 ;j ] ; issuchthat q [ i;j +1] = q [ i +1 ;j ] .Thenthesubgraphrepresenting x mustcontainasubgraphof 3 states q;r;s 2 Q suchthat q*r;q r;r*s ,and r s Proof. Set r = q [ i;j +1] = q [ i +1 ;j ] .Then r isthesourceforbothhorizontalandvertical transitionshavingtargetinstate s = q [ i +1 ;j +1] ,andlikewise, r istargetforbothhorizontal andverticaltransitionshavingsourceinstate q = q [ i;j ] 75

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Corollary2.3.4canbeappliedtoblockpathsinthesamemann er. Proposition4.2.6 Let besomegrid-innitepathin M F ( X ) =( Q;E;s;t; ) representingthepoint x 2 X havingdoubleperiod ( a;b ) for a;b 2 ,andlet 0 beany 2 2 block pathof where 0 = q [ i;j +1] *q [ i +1 ;j +1] q [ i;j ] *q [ i +1 ;j ] ; issuchthat q [ i;j ] = q [ i +1 ;j +1] .Thenthesubgraphrepresenting x mustcontainasubgraphof 3 states q;r;s 2 Q suchthat q*r;r q;r*s ,and s r Notethatwhen a = b =2,iftheconditionsofbothProposition4.2.5andProposit ion 4.2.6appearinthesame2 2blockpath,thentheresultingsubgraphisthatdepictedin Figure4.4(a).Forexample,graph(a)ofFigure4.4thatrepr esentsapointofdoubleperiod (2 ; 2)iscontainedasasubgraphofthegraphdepictedinFigure4 .5(c)thatrepresentsa pointofdoubleperiod(3 ; 2). Doublyperiodicpointshavingperiod(3 ; 3)canassumeseveraldierentgraphrepresentations.Asbefore,form G fromsome3 3factor 0 ofagrid-innitepaththat representssuchapoint.Thegraph G willcontainthree h -cyclesoflength3,foratotalof9 states(notallofwhichneedbedistinct).However,duetoth eimplicationsofProposition 2.3.1,weseethateach h -cycleoflength3iseithercomposedofthreedistinctstate s(a horizontalgraphtriangle)oristhesamestaterepeatedthr eetimes(an h -loop). Itseemsfeasiblethatasubgraphofminimumsizerepresenti ngapoint x ofleast doubleperiod(3 ; 3)mightbeobtainedbyusingstateamalgamationtocreate h -loops and/or v -loopsinthegraphrepresenting x .Sosupposetheedgesetofthesubgraph representingapointofperiod(3 ; 3)contains h -loopsand/or v -loops.Suchasubgraph couldbeconstructedfrom G bycollapsingoneormoreofthecyclesintoasinglestate. Forexample,ifwecollapseoneofthe h -cyclesintoasinglestate,thentheresulting subgraphisavalidsubgraphover7stateswithexactlyone h -loop.Inasimilarfashion, oneofthe v -cyclescouldbecollapsed.However,ifweattempttocollap setwoofthe h -cycles,aforbiddengraphdiamondwouldbecreated,whilec ollapsingallthree h -cycles wouldyieldtheuninterestingpointhavingperiod(1 ; 3).Itispossible,though,toachieve avalidsubgraphbycollapsingoneofthe h -cyclesandoneofthe v -cycles(thechoiceof 76

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whichisirrelevant,duetosymmetry).Theresultingsubgra phwillcontainasinglestate thathasboth h -loopsand v -loops.Suchasubgraphof5statesappearsinthegraphin Figure2.9asdoesasubgraphofsize5representingadoublyp eriodicpointofperiod (3 ; 2),butthetwosubgraphshavedieringedgesets.Inparticu lar,wehaveseenthat agraphrepresentingapointofperiod(3 ; 2)cannothaveanedgesetcontainingboth h -loopsand v -loops. However,thegraphofsize5discussedaboveneednotbethesm allestgraphcapableof representingapointofleastdoubleperiod(3 ; 3).Toexplorefurtherpossibilities,suppose theedgesetofthesubgraphrepresentingapointofperiod(3 ; 3)containsneither h loopsnor v -loops,andletusattempttoconstructsuchagraphfrom G throughstate amalgamation.WecanapplyPropositions4.2.5and4.2.6toa blockpathrepresentinga pointofdoubleperiod(3 ; 3)ifwearecarefulwithrespecttotheintersectionofhoriz ontal (vertical)graphtrianglesthatrestrictthesetofvalidla belsthatmayappearonthestates involved.Toeasefurtherdiscussion,letusrefertothesta teswithinthe3 3blockpath 0 numerically.Thatis,say 0 = 1 2 3 4 5 6 7 8 9 ; andletgraph G correspondtothisblockpath. Nowsupposeweamalgamatestate1withstate5sothatthecorr espondinggraph trianglesintersectatasinglestate.(Denotethissingles tatewith1 = 5asbefore.)Theresultingsubgraphwillthencontainapairofverticalgrapht rianglesandapairofhorizontal graphtrianglesintersectingwithoppositeorientationso nthe h -cyclesand v -cyclesthat passthroughthenewlyformed1 = 5state.(Seegraph(a)inFigure4.6:asdepicted,the h -cyclespassingthroughthe1 = 5stateareorientedcounterclockwise,whilethe v -cycles passingthroughthe1 = 5stateareorientedclockwise.)Alternatively,wecouldam algamate states5and7sothattheresultingsubgraphwillcontainapa irofverticaltrianglesand apairofhorizontaltrianglesintersectingwiththesameor ientationonthe h -cyclesand v -cyclesthatpassthroughthenewlyformed5 = 7state.(Seegraph(b)inFigure4.6.)We applyProposition4.2.5andProposition4.2.6tondvalidl abelsforthestatesofsuch 77

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(a) (b) b a a a b a b a a a a b b b a b a a a a a b b a a b a a a a b a b b a b a b a b b a b b b a a a b b b b a b b a b b b a a b b b 1/52 3 4 6 78 9 5/78 9 6 4 3 12 Figure4.6:Non-isomorphicgraphsrepresentingpointsofp eriod(3 ; 3) subgraphs:Forexample,thetwonon-isomorphicsubgraphsd epictedinFigure4.6both appearinthegraphrepresentingtheshiftofnitetypeden edbythesetof2 2blocks populatedwithsymbols a and b accordingtotherestrictionthateachblockcontainat leastone a .Otheramalgamationsinvolvingasinglepairofstatesaree itherforbiddenor isomorphictooneoftheunderlyinggraphsinFigure4.6.(Fo rexample,amalgamationof state5withstate q 2f 2 ; 4 ; 6 ; 8 g wouldcreateagraphdiamond;amalgamationofstates 5and3createsasubgraphisomorphictothatofgraph(b)inFi gure4.6;andsoon.) 1/5 6/7 b b a a b a a b a a b b b b b a b b a b a a b a a b a b 4 2 3 8 9 Figure4.7:Amalgamationoftwopairsofstates Wenextconsiderpairsofamalgamationsinvolvingpairsofs tates.Wecanapply Proposition4.2.5andProposition4.2.6totwopairsofstat esandtherebycreateasubgraph ofsize7thatisnon-isomorphictothepreviously-createds ubgraphofsize7thatcontained 78

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aloop.(SeetheFull-squareSystemgraphedinFigure3.6for suchasubgraph:Remove thestatelabeledwithallzeroes,removealledgesassociat edwiththatstate,andremove anyremaining h -loopsor v -loops.) Otherpossibleamalgamationsinvolvingtwopairsofstates fallintothreedistinct cases.Usingpropertiesofsymmetry,wemaywithoutlossofg eneralityrstformstate 1 = 5andtheninspectallotherpossibleamalgamationpairstha tmayoccurwithinthe samesubgraph. i)Theadditionalamalgamationof2 = 6or3 = 4wouldcreateahorizontalgraphdiamond(forexample,2 = 6 3 1 = 5and2 = 6 4 1 = 5),whiletheadditional amalgamationof4 = 8or2 = 7wouldcreateaverticalgraphdiamond. ii)Theadditionalamalgamationof6 = 7or3 = 8wouldlinkthethreehorizontalgraph trianglesinawaythatwouldmaintainconstantentriesonth emaindiagonalsofve ofthesevenstatesinthesubgraph.Thisallowstransitions totravelthroughthe amalgamatedstateswithoutforcingadditionallabelsonne arbystates.SeeFigure 4.7foranexampleofasubgraphwithvalidlabels. iii)Incontrasttocase(ii),theadditionalamalgamationo f2 = 9 ; 4 = 9 ; 6 = 8or3 = 7would linkthethreehorizontalgraphtrianglesinawaythatwould forceadditionallabels onotherstates,therebyreducingthesizeofthegraph.SeeE xample4.2.7forthe caseinvolvingtheamalgamationofstates6and8. 1/5 6/8 a b a a a a a a b a a a a a a b a b a a a a b b 4 3 9 7 Figure4.8:Subgraphshowingforcedlabels Example4.2.7 RefertoFigure4.8forasubgraphofthegraphinquestion.He re,labels ontheshadedstatesaredictatedbyCorollary2.3.6.Otherl abelsareforcedbyrst 79

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followingtheverticalgraphtriangle1 = 5 7 4,thenfollowingthehorizontalgraph triangle6 = 8 9 7,andnallyfollowingtheverticaltriangle6 = 8 3 9.Theresultis thatstates3and7areforcedtobearthesamelabel.Thisfurt herreducesthesizeofthe graphsincestates3and7mustbeamalgamated.Theresulting graphofsixstateswould havelabelsasprovidedinFigure4.9. 1/5 6/8 a a a a b a a a a a a b a b a a a a b b 4 3/7 9 a a b a 2 Figure4.9:Amalgamationofthreepairsofstates Insearchofagraphofminimumsizecapableofrepresentings omepointofleastdouble period(3 ; 3),theonlyotherinquiryistocheckwhetherthreestatesfr omthreedierent cyclesmightbeamalgamated.Withoutlossofgenerality,co nsidertheamalgamationof states1 ; 5,and9intoasinglestate.(Theamalgamationofstates1and 5withanyother statewouldimplythecollapseofacycle.)Infact,avalidsu bgraphdoesresultfromthe sequenceofamalgamationssuggestedbyinitiallyamalgama tingstates1 ; 5 ; and9. 1/5/9 78 2 Figure4.10:Amalgamationofstates1 ; 5,and9dictatesotherlabels i)States2and7maydieronlyintheupper-rightquadrant(R efertothesubgraphin Figure4.10,whichexcludesstates3 ; 4,and6sincethesestatesarenotpertinent.) State8musthavematchingentriesalongitsdiagonal.(seeP roposition2.3.3.)Therefore,allentriesintheshadedboxesmustagree,whichimpli esthatstates2and7 musthavethesamelabels. 80

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ii)Bythesameargumentasthatfoundinstepi),itcanbedete rminedthatstates6 and7havethesamelabel.Therefore,states2 ; 6,and7mustbeamalgamated.(See graph(a)ofFigure4.11.) iii)Toavoidtheexistenceofhorizontalgraphdiamondsbet weenstates1 = 5 = 9and2 = 6 = 7, states3 ; 4,and8mustbeamalgamated.(Seegraph(b)ofFigure4.11,wh ereone setofvalidlabelsissupplied.) 1/5/9 2/6/7 348 (a) (b) b b a b a a b b b b b a Figure4.11:Multipleamalgamationssuggestedby1 = 5 = 9amalgamation Soithasbeendemonstratedthat3statesistheminimumnumbe rrequiredforagraph torepresentthepointhavingleastdoubleperiod(3 ; 3).Thereare,however,twononisomorphicgraphsofsize3capableofthis:thatis,wecould amalgamatethesetsofstates thatappearalongthecounter-diagonalintheblockpathrep resenting G instead.This wouldcreatea3 = 5 = 7state,a1 = 6 = 8state,anda2 = 4 = 9state.Suchasubgraphofsize3 canbefoundintheThree-dotSystem,representingthepoint ... ... ... ... ... ... 101101 011011 110110 ... ... ... ... ... ... (4.2.2) ThesegraphtrianglesintheautomatonrepresentingtheThr ee-dotSystemtaketheappearanceof h -cyclesand v -cyclesthatexhibitoppositeorientationsasdictatedbyC orollary2.3.4.Inthiscase,thereareconstantentriesalongth ecounterdiagonalsofpointsin theshiftspace.Incontrast,thegraphofsize3depictedind iagram(b)ofFigure4.11has h -cyclesand v -cycleswiththesameorientation,whichproducesconstant entriesalongthe 81

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maindiagonalsofpointsintheshiftspace. Overthecourseofexaminingpointshavingleastdoubleperi od( a;b )for a;b 2f 1 ; 2 ; 3 g itwasdeterminedthattheonlydoublyperiodicpointsofthi stypethatcouldberepresentedbyagraphofsize3werethepoint(3 ; 3)andtheuninterestingpoints(1 ; 3)and (3 ; 1).Wecangeneralizedoublyperiodicpointswithrespectto graphsofsize3bythe inclusionofoneadditionalgraph.Example4.2.8 Considerthegraph G ofsize3giveninFigure4.12.(Suchagraph appearsasasubgraphofthegraphinFigure3.6representing theFull-squareSystem.) 0 0 0 0 1 0 0 1 0 1 1 0 Figure4.12:Representedpointshaveperiod(1 ;n )forall n 3 Continuouslyfollowingtheverticalgraphtriangleof G describesapointintheshift space X havingleastdoubleperiod(1 ; 3).However,otherpointsofleastdoubleperiod (1 ; 3+ n )aredescribedbyrepetitionsofanextended v -cyclethatresultsfromtheinclusion of n tripsaroundthe v -looplocatedatthestateofall0's.Forexample,thefollow ing pointofleastdoubleperiod(1 ; 4)resultswhentripsaroundtheverticalgraphtriangleare interruptedwithsingletripsaroundthe v -loop. ... ... ... 111 000 000 111 000 000 ... ... ... Naturally,Figure4.12andExample4.2.8couldbereworkedi ntermsofhorizontal 82

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graphtrianglesandleastdoubleperiods( m; 1)for m 3.This,alongwithLemma4.2.2, veriesthefollowing.Proposition4.2.9 Suppose x isadoublyperiodicpointofthetwo-dimensionalshiftof nitetype X .If x canberepresentedbythesubgraph G of M F ( X ) having j G j =3 ,then x hasleastperiod ( a;b ) 2f (1 ;n ): n 3 g[f ( n; 1): n 3 g[f (3 ; 3) g Two-dimensionalpointsofthetypeexpressedby(4.2.2)can beviewedasinnite copiesofaone-dimensionalpointhavingleastperiod a whereeachrowisacopyofthe rowdirectlybelowitthathasbeenshifteddiagonallytothe rightonespace.Thatis,for apoint x intheshiftspace, x ( i;j ) = x ( i +1 ;j +1) 8 ( i;j ) 2 Z 2 .Agreatvarietyofthesetypesof doublyperiodicpointsarenaturallyfoundintheDiagonalshiftSystem. Example4.2.10 Considerthegraph G ofsize4giveninFigure4.13.(Suchagraph appearsasasubgraphofthegraphinFigure2.7representing theDiagonal-shiftSystem.) 0 0 0 0 0 0 0 1 0 1 0 0 1 0 10 1 q 0 q 3 q 4 q Figure4.13:Representedpointshaveperiod( n;n )forall n 4 Thesubgraph G iscapableofrepresentingpointshavingleastdoubleperio d( a;a )for all a 4.Forexample,thepointofleastdoubleperiod(4 ; 4)expressedin(4.2.3)results whentripsaroundthe h -cycle( v -cycle)oflength4arerepeatedwithouttheinclusionof the h -loops( v -loops)atthestateofall0's. Theperiodicpointin(4.2.3)isexpressedthroughrepetiti onofthe4 4blockhaving 1'salongthediagonaland0'selsewhere.Otherpointsoflea stdoubleperiod(4+ n; 4+ n ) aredescribedbyrepetitionsofanextended h -cycle( v -cycle)thatresultsfromtheinclusion of n tripsaroundthe h -loop( v -loop)locatedatthestateofall0's.Thesepointsofleast doubleperiod(4+ n; 4+ n )areexpressedthroughrepetitionofthe(4+ n ) (4+ n )block having1'salongthediagonaland0'selsewhere. 83

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... ... ... ... ... ... ... ... 00010001 00100010 01000100 10001000 00010001 00100010 01000100 10001000 ... ... ... ... ... ... ... ... (4.2.3) Proposition4.2.11 Considerthesetofallshiftsofnitetypehavingtheproper tythat F ( X )= A ( X ) .Thenfor a 2f 1 ; 2 ; 3 g ,theminimumsizeofagraphcapableofrepresenting somepointofleastdoubleperiod ( a;a ) is a ,andfor a 4 ,theminimumsizeofagraph capableofrepresentingsomepointofleastdoubleperiod ( a;a ) is 4 84

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5MonoidsandFollowerSets Toeachone-dimensionallanguage,onemayassociatea monoid (asetwithanassociative binaryoperationandanidentity)calledthesyntacticmono idofthelanguage,whichisnitepreciselywhenthelanguageisrecognizable.Furtherm ore,witheachone-dimensional recognizablelanguageonemayassociateaniteautomatono fminimalsizeandacorrespondingtransitionmonoidthattakesthewordsofthelan guageasasetoffunctions actingontheautomaton.Foraone-dimensionallanguagetha tisrecognizable,itisknown thatthesyntacticmonoidofthelanguageisisomorphictoth etransitionmonoidofits minimalautomaton.(See,forexample,[24].)Inone-dimens ionalsymbolicdynamics,a similarnotionisthatoffollowersets,whicharetheinnit esetsofwordsthatcanfollow anygivenwordinthelanguageoftheshiftspace.(Intheonedimensionalcase,follower setsarealwaysinnitesincewordsintheallowedlanguageo faone-dimensionalshift spacearealwaysprolongableandthereforewordscanalways beextendedindenitely.)A one-dimensionalshiftspaceissoc(recognizable)ifando nlyifithasanitenumberof followersets[25].Inalabeledgraphrecognizingasocshi ftspace,thefollowersetofa stateisthereforedenedtobethecollectionoflabelsofpa thsoriginatingatthatstate. Inthischapter,weintroducetheideaofmonoidsandfollowe rsetsinthetwo-dimensional case.Thisallowsustoinvestigatenotionsofequivalencei ntheblocksofatwo-dimensional languageandinthestatesofagraphrepresentingatwo-dime nsionalshiftspace.InSection5 : 1wedenemonoidsbasedontheallowedconcatenationsofblo ckshavingthesame height.Boundsarethenplacedonthesizeofthesemonoidsfo rcertaindotsystems.In Section5 : 2wegeneralizethenotionofone-dimensionalfollowersets to M F ( X ) graphsthat representtwo-dimensionalshiftspaces.Example5.2.4rev iewsallthekeyelementsofthis dissertation. 85

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5.1MonoidsforDotSystems Indeningabinaryoperationfortwo-dimensionalblocks,t heimmediatedicultyisone ofclosure.Ifwedesirethe\product"oftwoblockstoyielda notherblock(inthesenseof concatenation,tobemademorepreciseinamoment),wemustr eturntotheconvention ofinspectingthehorizontalcaseandtheverticalcasesepa rately,sincetheproductof twoblockshavingdierentheightscouldproduceanon-bloc kshape.Someeorthas beenmadetodeneatypeofdiagonalproductproducingnon-b lockshapes,butsuch researchhasbeenlimitedtoone-letteralphabets[1].Tota kefulladvantageoftherich structureinherentintwo-dimensionallanguages,wewilll ookmorecloselyatthedenition offollowersetsinthenextsection.Here,wementionsomere sultsachievedbylimiting thediscussiontoblocksofauniformheight. Twoblocksofthesameheightmaybe horizontallyconcatenated toformanewblock. Wedenotethebinaryoperationby B m;n 1 B 0 m;n 2 anddenetheresultingblockby B m;n 1 + n 2 ( i;j )= 8<: B 0 m;n 1 ( i;j ):0 i n 1 1 ; 0 j m 1 B 00 m;n 2 ( i n 1 ;j ): n 1 i n 1 + n 2 1 ; 0 j m 1 : Informally,thesymbolsfor B 00 m;n 2 arecopiedtotherightofthesymbolsfor B 0 m;n 1 insuch awayastocreateanewblockhavingdimension m ( n 1 + n 2 ).Inthesamemanner,two blocksofthesamelengthmaybe verticallyconcatenated .Thatbinaryoperationwillbe denotedby B m 1 ;n B 0 m 2 ;n withtheresultingblockdenedby B m 1 + m 2 ;n ( i;j )= 8<: B 0 m 1 ;n ( i;j ):0 i n 1 ; 0 j m 1 1 B 00 m 2 ;n ( i;j m 1 ):0 i n 1 ;m 1 j m 1 + m 2 1 : Informally,thesymbolsfor B 00 m 2 ;n arecopiedabovethesymbolsof B 0 m 1 ;n insuchawayas tocreateanewblockhavingdimension( m 1 + m 2 ) n Toeachpicturelanguage L ,wemayassociatesemigroupsbasedontheallowedconcatenationsbetweenblocks.(Asemigroupisasetwithabina ryassociativeoperation.) Consider R m = f m;n : n 0 g ,i.e.thesetofallblockshaving m rows;Inparticular, theemptyblock isanelementof R m forall m sincewecanview ashavingdimension m 0.Wedene horizontalsyntacticequivalence h on R m relativeto L asfollows.Given 86

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1 ; 2 2 R m ,saythat 1 h 2 i 8 i ; j 2 R m i 1 j 2 L i 2 j 2 L .Equivalenceclassesshallbedenotedby[ ]andtheircontextsby ([ ]),e.g.if 1 2 2 L ,then wewoulddenotethisby( 1 ; 2 ) 2 ([ ]).Rightandleftcontextsaredenedintheobvious way.The m -horizontalsyntacticsemigroup of L withtheoperation[ 1 ] [ 2 ]:=[ 1 2 ] isdenoted H m ( L )= R m = h .Notethatunless m; L ,the m -horizontalsyntactic semigroupof L willalwayscontainanelementrepresentingtheclassofblo cksthatarenot inthelanguage;wedenotethis zero by 0 = f : 2 R m ; 62 F ( X ) g .Moreimportantly, forall m ,thesemigroup H m ( L )willcontainthe identityelement e =[ ].(Foranyblock 1 2 R m ,[ 1 ] [ ]=[ 1 ]=[ 1 ]and[ ] [ 1 ]=[ 1 ]=[ 1 ].)Soinfact, H m ( L )is amonoid. Verticalsemigroupsmaybedenedon C n = f m;n : m 0 g ,thesetofallblocks having n columns,inasimilarmanner.Dene verticalsyntacticequivalence v on C n relativeto L asfollowsGiven 1 ; 2 2 C n ,saythat 1 v 2 i 8 i ; j 2 C n i 1 j 2 L i 2 j 2 L .The n -verticalsyntacticsemigroup of L withtheoperation [ 1 ] [ 2 ]:=[ 1 2 ]canbedenoted V n ( L )= C n = v .However,weshalldiscuss horizontalsyntacticequivalenceonly,asverticalsyntac ticequivalenceisanalogous. Nowsuppose X isadotsystemdenedthroughsome r c shape S .Foreaseofnotation, whendiscussingthefactorlanguageofadotsystemwedenote H m ( F ( X ))= H m ( A ( X )) bysimply H m .For m
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behorizontallyconcatenatedwithinthelanguage F ( X ).Thatis,if( 1 ; 2 ) 2 ( e ),then 1 and 2 meetindirection(1 ; 0)atdistance0within F ( X ).Unlikethedotsystemsdened byone-columnshapesasinExample5.1.1,formostdotsystem shorizontal(andvertical) translationsofthedeningshape S dictatewhetherblocksmaymeetindirection(1 ; 0)and ifsoatwhatdistance.Thisaectsthesizeofthehorizontal syntacticmonoids.Itshould benotedthatthezero 0 representingconcatenationsinvolvingblocks B 2 R m n F ( X )also representspairsofblocksfromthelanguagethatdonotmeet indirection(1 ; 0)within F ( X ).Soif 1 ; 2 2 F ( X ) \ R m and 1 doesnotmeet 2 indirection(1 ; 0)within F ( X ), then( 1 ; 2 ) 2 ( 0 ). Todeterminethesizeofthe r -horizontalsyntacticmonoid H r ,itmustbedetermined howmanyhorizontaltranslationsof S willaect j H r j sincethereisnoboundonthelength oftheblocksthatmayrepresenttheequivalenceclasses.Fo ragivenshape,wetherefore inspectblocksoflength n =1 ; 2 ;::: tond\good"representativesfortheequivalence classesof H r .Letusrstinspecttwoexampleswherehorizontaltranslat esof S donot aectthesizeof H r regardlessofthelengthoftheblocksusedtorepresentthee quivalence classes.Thismakesitpossibletoeasilydeterminethesize ofboth H r and H m Example5.1.2 Supposethedotsystem X isdenedbythe1 2shape S = f (0 ; 0) ; (1 ; 0) g Anexampleofapointcontainedinthisshiftspaceis: ... ... ... ... 1111 0000 1111 1111 ... ... ... ... Forablock B m;n 2 F ( X ),allhorizontaltranslationsof S within B mustmaintainasum equivalenttozero.Thisrequiresthatone-rowdesignsbepo pulatedwiththesamesymbols asthosefoundintheinitialcolumnofanygivenblock;thati s, 8 j 2f 0 ; 1 ;:::;m 1 g ,either B m;n ( i;j ) 7!f 0 g forall i 2f 0 ; 1 ;:::;n 1 g or B m;n ( i;j ) 7!f 1 g forall i 2f 0 ; 1 ;:::;n 1 g Ontheotherhand,forone-columndesignsthereisnorestric tiononthestringofsymbols thatmayappearsinceaccordingtoCorollary1.3.2, j F m; 1 ( X ) j =2 m .Therefore,the 88

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set F m; 1 ( X )canbeusedtorepresent2 m distinctequivalenceclassesfor H m :thatis,for B m; 1 ;B 0 m; 1 2 F m; 1 ( X ), B m; 1 h B 0 m; 1 i B m; 1 = B 0 m; 1 .Soincluding 0 and e j H m j =2+2 m Example5.1.3generalizesExample5.1.2toarectangularsh apecomprisedoftworows. Forthisshape,onemustconsiderthesumofthesymbols(rath erthanjustasinglesymbol) withineachcolumnof S .Althoughthesymbolsappearinginaone-rowdesignoflengt h n neednotbeuniformasinExample5.1.2,thecolumnsthatmaya ppearinhorizontal translatesof S arerestrictedbyawallpaperpattern. Example5.1.3 Recallthatthe2 2shape S = f (0 ; 0) ; (1 ; 0) ; (0 ; 1) ; (1 ; 1) g denesthe Full-squareSystem X .Anexampleofapointcontainedinthisshiftspacewouldbe: ... ... ... ... 1001 0110 1001 1001 ... ... ... ... If B 2 F 2 ;n ( X ),thentakingthesumoveranycolumnof B mustyieldthesameresult: thatis,either B ( i; 0)+ B ( i; 1) 0mod2forall i 2f 0 ; 1 ;:::;n 1 g or B ( i; 0)+ B ( i; 1) 1mod2forall i 2f 0 ; 1 ;:::;n 1 g .Let T = f (0 ; 0) ; (1 ; 0) g anddenethetwoone-column designs 0 : T 7!f 0 g and 1 : T !f 0 ; 1 g with 1 (0 ; 0)=0and 1 (1 ; 0)=1.Notethat both( 0 ; 0 )and( 1 ; 1 )arein ( e ).However,( 0 ; 0 ) 2 ([ 0 ])but( 1 ; 1 ) = 2 ([ 0 ]) whereas( 1 ; 1 ) 2 ([ 1 ])but( 0 ; 0 ) = 2 ([ 1 ]).Therefore H 2 canberepresentedby [ 0 ] ; [ 1 ], e ,and 0 sothat j H 2 j =2+2 1 Forthesizeofthe m -horizontalsyntacticmonoid H m when m isallowedtoexceed r =2,rstconsiderone-columndesignsofheight m andtheequivalenceclassesthatthey represent.Inthiscase,therewillbe m 2verticaltranslationsoftheshape S alongany one-columndesign B 2 R m .Theoriginal(normalized)shape S andthe m 2vertical translationsof S thatintersecttheone-columnblock B willthereforeproducea(vertical) sequenceoflength m r +1= m 2+1= m 1overthealphabet= f 0 ; 1 g assums aretakenwithintheinitialcolumnoftranslatesof S .ByCorollary1.3.2therewillbe norestrictiononthestringofsymbolsthatmayappearinaon e-columndesignsothat 89

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2 m 1 distinctsequencesareproduced.Furthermore,duetothela ckofafreecellinthe two-columnshape S ,thissequenceofsumsdictatesthesequenceofsumsforallo ther columnsappearinginanyhorizontalblockextensionof B .(Forthe2 2shape S ,there areonlytwowallpaperpatterns:eitherthesumofthebitsof eachcolumnisevenorthe sumofthebitsineachcolumnisodd.)Therefore,blocksinth eleftandrightcontextof block B mustexhibitthesamepatternofcolumnarsumsaseachothera ndas B itself. Notethat e isadistinctclasssinceallone-columnblocksofheight m areintheright contextof e ;thatis,( ";B ) 2 ( e )forall B 2 F m; 1 ( X )= f 0 ; 1 g m; 1 .Sowiththeexception of e and 0 ,thereexistsaone-to-onecorrespondencebetweenthese2 m 1 sequencesand equivalenceclassesof H m .Therefore,for m 2, j H m j =2+2 m 1 NotethatinExample5.1.3,allone-columnblocksareinther ightcontextof e .However,whenaone-columnblockisusedtorepresentanequival enceclass,itdiscriminates betweenotherone-columnblocksthatmaybecontainedinits rightcontext.Inthisway, thelength c oftheshape S aectsthesizeofthe r -horizontalsyntacticmonoidforall r c shapes,sinceblocksoflength n
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B 0 r;n 0 h B r;n ,thenitmustbethecasethat n 0 n inorderfor n 0 + n + c forall Lemma5.1.4suggestsalowerboundonthesizeofthe r -horizontalsyntacticmonoid foranydotsystem.Theboundisbasedontheneedfordistinct equivalenceclassesfor blocksoflength n c 1. Proposition5.1.5 If X isadotsystemdenedthroughan r c shape S ,then j H r j 2+2( c 1) Proof. Twoelementsof H r alwaysexistandarealwaysdistinct:thezero 0 always exists(thefullshiftdoesnotexistfordotsystemsdenedb yniteshapes S )andis alwaysdistinct(ablock B 2 R m n F ( X )cannotbelongtotherightcontextofanyother element);andtheidentity e alwaysexists(wecanalwaysconcatenatezeroblocksofany height)andisalwaysdistinct( e istheonlyelementof H r tocontainallone-column blocksinitsrightcontextunless S hasonlyonecolumnasinExample5.1.1,inwhich caseExample5.1.1providesthat j H 2 j =2 2+2( c 1)since c =1).Lemma5.1.4 indicatesthatblocksoflength c 1maynotbelongtoanyequivalenceclass[ B r;n ]having 0
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monoidsincetheymaydisruptthesymmetryoftheshapeand/o rmaycontributetothe dotsystemhavingfactorlanguagethatishorizontallytran sitive.Forexample,the2 2 diagonalshapedenesadotsystemhaving m -horizontalsyntacticmonoidlargerthan thatofthe2 2rectangular(square)shape. Example5.1.6 Let X betheDiagonal-shiftSystemdenedviathe2 2shape S = f (0 ; 0) ; (1 ; 1) g .Anexampleofapointcontainedinthisshiftspacewouldbe: ... ... ... ... 1011 0110 1101 1011 ... ... ... ... Pointsinthisshiftspacecanbeplacedinaone-to-onecorre spondencewithpointsinthe one-dimensionalfull2-shift:Firstmapanybi-inniteseq uence :::x 1 x 0 x 1 ::: ontothe x -axisaccordingtothemap P : x i 7! ( x i ; 0)for i 2 Z ;thenmapshiftsofthebi-innite sequenceindiscretetimestepsaccordingtothemap P t : x i 7! ( x i + t ;t )for t 2 Z .Note thatsinceanytwoblocksmaybeconcatenatedinthelanguage oftheone-dimensionalfull 2-shift,theone-dimensionalfull2-shifthasonlyoneelem ent,theidentity,initssyntactic monoid.Wewilldemonstratethatinthetwo-dimensionalcas e,thisshapecreatesan m -horizontalsyntacticmonoidofsubstantialsize. Denethefourblocks i : f (0 ; 0) ; (0 ; 1) g!f 0 ; 1 g by 0 (0 ; 0)=0 ;; 0 (0 ; 1)=0 1 (0 ; 0)=1 ;; 1 (0 ; 1)=0 2 (0 ; 0)=0 ;; 2 (0 ; 1)=1 3 (0 ; 0)=1 ;; 3 (0 ; 1)=1 Eachofthesefourblocksrepresentsadistinctequivalence classin H 2 .Forexample,the readercaneasilyverifythatwhile 0 and 2 havethesamerightcontext,theirleftcontexts 92

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dier;ontheotherhand, 1 and 3 havethesamerightcontext(whichdiersfromthatof 0 and 2 ),buttheirleftcontextsdier.Otherthan e and 0 ,therearenootherequivalence classesfor j H 2 j ,sinceforall n 1,( B 00 2 ;n 00 ;B 0 2 ;n 0 ) 2 ([ B 2 ;n ])i B 00 2 ;n 00 ( n 00 1 ; 0)= B 2 ;n (0 ; 1) and B 2 ;n ( n 1 ; 0)= B 0 2 ;n 0 (0 ; 1).Therefore, j H 2 j =2+2 1 2 1 Nowconsiderthesizeofthe m -horizontalsyntacticmonoid H m when m isallowed toexceed r =2.UnlikeExample5.1.3,given B m;n 2 F ( X )representingaequivalence classin H m ,thecolumnsofblock B m;n appearingwithinhorizontaltranslatesof S need notcarryanyparticularpatternofsums.However,blocksin thefactorlanguagedo exhibitadiscerniblepatternthatmustcyclethroughacert ainnumberofcolumns.For example,ifweconsiderthreeblocks B m;n ;B 0 m;n 0 ;B 00 m;n 00 2 F ( X ) \ R m ,then( ";B 0 m;n 0 ) 2 ([ B m;n ])ifor j 2f 0 ; 1 ;:::;m 2 g thecells B 0 m;n 0 ( n 0 1 ;j )= B m;n (0 ;j +1),and ( B 00 m;n 00 ;" ) 2 ([ B m;n ])ifor j 2f 0 ; 1 ;:::;m 2 g thecells B m;n ( n 1 ;j )= B 00 m;n 00 (0 ;j +1). Soforanytwoblocks, B m;n h B 0 m;n 0 ifandonlyif B m;n ( n 1 ;j )= B 0 m;n 0 ( n 0 1 ;j )for j 2f 0 ; 1 ;:::;m 2 g and B m;n (0 ;j )= B 0 m;n 0 (0 ;j )for j 2f 1 ; 2 ;:::;m 1 g .Thereforethere are2 m 1 dierentone-columndesigns L correspondingtodistinctequivalenceclasses [ L ]accordingto( ";B 0 m;n 0 ) 2 ([ L ])andthereare2 m 1 dierentone-columndesigns R correspondingtodistinctequivalenceclasses[ R ]accordingto( B 00 m;n 00 ;" ) 2 ([ R ]).Since F ( X )ishorizontallytransitive,foreveryorderedpairofonecolumnblocks( L ; R ),there existsablockoflength n = m +1suchthatfor0 j m 1, L = B m;m +1 (0 ;j )and R = B m;m +1 ( m;j ).Thatis,sincetheinitialcolumnof B m;m +1 aectsonly B m;m +1 ( i;j )= B m;m +1 ( i +1 ;j +1)for0 i m 1and i j m 1,thedesignoftheinitial(far-left) columnof B m;m +1 isindependentofthedesignofthenal(far-right)columno f B m;m +1 (Theinitialcolumnonlyaectsthecellsinandabovethecou nterdiagonal.)Therefore, theblocksoflength m +1represent2 m 1 2 m 1 distinctequivalenceclasses.Furthermore, anyblockoflength n 1canplacedintooneoftheseequivalenceclassesbasedonit s initialandnalcolumns.Forexample,since F ( X )ishorizontallytransitive,thereexists someblockoflength m +1wherethedesignoftheinitialcolumnisthesameasthatof thenalcolumn,sothatsuchblocksareequivalenttotheset ofone-columndesigns.So for m 3 ; j H m j =2+(2 m 1 ) 2 Considerthe translationsequence x = x 1 ;x 2 ;:::x c + n 1 ofevenandoddsumstaken overthecellsthatlieinthe c + n 1distincttranslatesof S thatintersectblocksofheight 93

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r andlength n .Forthediagonalshape S = f (0 ; 0) ; (1 ; 1) g ,horizontaltranslatesofthe shape S intersectpairsof(diagonal)cellsinarepresentativeblo ckonceandonlyonce, sothatforthisshapeeachof2 c + n 1 sequencesisrealized.Proposition5.1.7extendsthis processto r r shapescomprisedofasinglediagonalthroughasquareregio n.Although r = c ,wedenoteby c thevaluesthatpertaintothelengthof S Proposition5.1.7 Suppose r = c 2 andlet X beadiagonal-shiftsystemdenedvia the r c shape S = f (0 ; 0) ; (1 ; 1) ;:::; ( r 1 ;c 1) g .Then j H r j =2+ c 2 X n =1 2 c + n 1 +(2 c 1 ) 2 : Proof. ByLemma5.1.4,forallshapesandforeach n 2f 1 ; 2 ;:::;c 2 g thereexistdistinct equivalenceclassesforblocksoflength n ,sinceonlyundersizedblockscanindiscriminately acceptotherundersizedblocksintheircontext.Inparticu lar,forthe r c diagonalshape, thereexists2 c + n 1 distinctequivalenceclassesforeach n 2f 1 ; 2 ;:::;c 2 g ,sincean equivalenceclassrepresentedbyanundersizedblock B r;n correspondstoasequenceof length c + n 1baseduponthesumofthebitsin B r;n thatallows( B 0 r;n 0 ;B 00 r;n 00 ) 2 ([ B r;n ]) whenever n 0 + n + n 00 = c .Thatis,eachof c + n 1horizontaltranslationsof S can potentiallyintersectadesignoflength n withasumthatiseitherevenorodd.(See Figure5.1.) . Figure5.1:Horizontaltranslationsof S Nowconsidertheequivalenceclassesforblocksoflength n c 1.Suchclassesare bestrepresentedbyblocksoflength c +1sinceforblocksofthislength,theinitialcolumn of B r;c +1 issuchthat B r;c +1 (0 ; 0)= i = r 1 X i =1 B r;c +1 ( i;i ) : 94

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Sinceonlythecellsonthecounterdiagonaloftheunderlyin g c c squareshapeareaected, thedesignoftheinitial(far-left)columnof B r;c +1 isindependentofthedesignofthenal (far-right)columnof B r;c +1 .Therefore,theblocksoflength c +1compriseexactly2 c 1 2 c 1 distinctequivalenceclassessinceblocksacceptedinthel eftcontextofaclassrepresented byablockoflength c +1areindependentofblocksacceptedintherightcontextof the class.Finally,anyblock oflength n = c 1, n = c or n>c +1canbeplacedintoan equivalenceclassrepresentedbyablockoflength c +1thathasthesameinitialandnal columnsas Twoblocks B r;n ;B 0 r;n 0 areequivalentinthe r -horizontalsyntacticmonoidonlyifthey bearthesamesumforallhorizontaltranslatesof S .Theprocessofcountingdistinct translationsequencesinordertodeterminethesizeofthe r -horizontalsyntacticmonoid canbeextendedtoanyshape,regardlessoftheheight r oftheshape.Wecanplacean upperboundonthesizeof H r byassumingthatallhorizontaltranslationsequencesare achieved.Forexample,theDiagonal-shiftSystemprovides asharpupperboundfor j H r j relatedtodotsystemsdenedby r 2shapes.ThegeneraldiagonalsystemofProposition 5.1.7providesasharpupperboundforallothershapes.Corollary5.1.8 Let X beadotsystemdenedbyan r c shape S .Thenfor c 3 j H r j 2+ c 2 X n =1 2 c + n 1 +(2 c 1 ) 2 : Attheotherextremearedotsystemsdenedbyrectangularsh apeslackingfreecells. Forexample,letadotsystem X bedenedbyan r c rectangularshapeandconsider theundersizedblocksoflength n c 2asrepresentativesforanequivalenceclass.The conditionthattwoblocks B r;n ;B 0 r;n bearthesamesumforallhorizontaltranslatesof S whilenecessary,isnotsucientforthetwoblockstobeequi valentinthe r -horizontal syntacticmonoid.Forthepurposeofmatchingwallpaperpat terns,twoblocks B r;n ;B 0 r;n areequivalentin H r ifandonlyiftheybearthesamesumovereachcolumn.Forrefe rence, Lemma5.1.9restatesausefulresultfoundwithintheproofo fProposition3 : 1 : 2.(Refer toFigure3.1asneeded.)Lemma5.1.9 Let X beadotsystemdenedthroughsome r c rectangularshape S that 95

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lacksfreecells.Thenfor B r;n with n c B r;n 2 F ( X ) ,8 i 0 2f 1 ;:::;n c g i 0 + c 2 X i = i 0 r 1 X j =0 ( B r;n ( i;j ))= r 1 X j =0 ( B r;n ( i 0 1 ;j ))= r 1 X j =0 ( B r;n ( i 0 + c 1 ;j )) : Becauseofthesewallpaperpatterns,itispossibletodeter minetheexactsizeofthe m horizontalsyntacticmonoidsassociatedwithdotsystemsd enedbyrectangularshapes thatlackfreecells.Thecasewhen c =1isencompassedbyExample5.1.1,andall rectangularshapeshaving c =2areanalogoustoExample5.1.3.(Regardlessofthe height r oftheshape,when c =2thereareonlyfourequivalenceclassesfor H r :[ 0 ],in whosecontextallelementshavecolumnsthatsumto0(mod2); [ 1 ],inwhosecontext allelementshavecolumnsthatsumto1(mod2); e ,inwhosecontextboth( "; 0 )and ( "; 1 )appear;and 0 ,inwhosecontext( 0 ; 1 )appears.Forcaseswhere m r ,thereare twopossibilitiesforeachone-rowextensionsothat j H m j =2+2 m r +1 .)Forwallpaper patternsdenedbyrectangularshapesoflength c 3,weisolatethecase H r inLemma 5.1.10beforeweexaminethemoregeneralcase H m .Lemma5.1.10usesthenotionof primitivewords:A primitiveword isawordthatcannotbewrittenintheform u i forany word u andnumber i> 1.Sotheprimitivewordisnotapowerofanyotherword.(Here concatenationistakenasmultiplication.)Ifweuse D todenoteaone-columndesignwith bitsthatsumto1(mod2)and E todenoteaone-columndesignwithbitsthatsumto0 (mod2),thenwecanrefertowallpaperpatternsaswordsinao ne-dimensionallanguage overthealphabet= f D;E g Lemma5.1.10 For c 3 ,let X beadotsystemdenedthroughsome r c rectangular shape S thatlacksfreecells.Let = f d :1 d c 1 ;c 0 modd g ,andlet W bethe setrepresentingallwallpaperpatternsoflength c wherethewords w 1 w 2 w c = w 2 W aresuchthatfor i 2f 1 ; 2 ;:::;c g ,thesymbol w i 2f D;E g .Let W c W denotethe primitivewordsoflength c ,andfor d 2 lettheset W d W bethesetofallnonprimitivewordsthatmaybewrittenintheform w =( w 1 w 2 w d ) c=d .Then j H r j = 2+2 1 +2 2 + ::: +2 c 2 + c j W c j +( d 1) j W d j Proof. Weplacearbitraryblocksintoequivalenceclassesbasedup ontheirlength n .First consideranequivalenceclassrepresentedbyablock B r;n oflength0 n
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willndthatotherundersizedblocksofvaryinglengthsare groupedintotherightcontext basedsolelyontheirlengths;thatis,Lemma1.3.1guarante esthat( ";B 0 r;n 0 ) 2 ([ B r;n ]) whenever n 0 c n 1,regardlessofthedesign B 0 r;n 0 .Ontheotherhand,[ B r;n ]will alsocontainblocksoflength n 0 >c n 1initsrightcontext,buttheselongerblocks belongtotherightcontextofanequivalenceclassonlyifth eirwallpaperpatternagrees withthepartialpatternspeciedby B r;n .Forexample,if n 0 >c ,( ";B 0 r;n 0 ) 2 ([ B r;n ]) onlyifblock = B r;n B 0 r;n 0 hasthepropertythat P r 1 j =0 ( i;j )= P r 1 j =0 ( i + c;j )for all i 2f 0 ; 1 ;:::;n 1 g asinLemma5.1.9.Therefore,eachwallpaperpatternofleng th n
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bestrepresentedbyblocksoflength c 1 ;c;:::c 1+( d 1);thatis,eachnon-primitive word(wallpaperpattern)oflength c isrepresentedbyablockoflength c 1andthen creates( d 1)otherdistinctequivalenceclassesuntilitreachesablo ckoflength c 1+ d whichpermitsremovalofablockoflength d andstillleavesablockoflength c 1to representtheequivalenceclass.Sincethesewallpaperpat ternsrepeatevery d translates, blocksoflength n c + d 1permitremovalofablockof(somemultipleof)length d andstillleaveablockoflength n c 1.Inthesameway,aprimitiveword(wallpaper pattern)oflength c isrepresentedbyablockoflength c 1andthencreates c otherdistinct equivalenceclassesuntilitreachesablockoflength2 c 1whichpermitsremovalofa blockoflength c andstillleavesablockoflength c 1torepresenttheequivalenceclass. Sinceallpatternsincludingtheprimitivewallpaperpatte rnsrepeatevery c translates, blocksoflength n 2 c 1permitremovalofablockof(somemultipleof)length c and stillleaveablockoflength n c 1. Foran r c rectangularshape S ,ablock B m;n ofheight m>r stilldisplaysawallpaper patternoflength c ,asanysubblock B 0 r;n B m;n mustitselfdisplayawallpaperpatternof length c .Thismakesitpossibletodeterminetheexactsizeofthe m -horizontalsyntactic monoid.Todoso,weconsiderwords(wallpaperpatterns)for medfromanalphabetof one-columnblocks.Thealphabetrepresentsthepotentials equenceofsumsthatmayexist forverticaltranslatesof S withinarepresentativeblock. Proposition5.1.11 For c 3 ,let X beadotsystemdenedthroughsome r c rectangularshape S thatlacksfreecells.Let = f d :1 d c 1 ;c 0 modd g ,andlet W 0 bethe setrepresentingallwallpaperpatternsoflength c wherethewords w 0 1 w 0 2 w 0 c = w 0 2 W 0 aresuchthatfor i 2f 1 ; 2 ;:::;c g ,thesymbol w 0 i 2f D;E g m r +1 ; 1 isaone-columnblock. Let W 0 c W 0 denotetheprimitivewordsoflength c ,andfor d 2 lettheset W 0 d W 0 be thesetofallnon-primitivewordsthatmaybewritteninthef orm w 0 =( w 0 1 w 0 2 w 0 d ) c=d Thenfor m r j H m j =2+(2 1 ) m r +1 +(2 2 ) m r +1 + ::: +(2 c 2 ) m r +1 + c j W 0 c j +( d 1) j W 0 d j Proof. Theproofisbyinductionon m r .Thecase m = r isshowninLemma 5.1.10.Nowassumetheresultistruefor m>r .Wewanttoshowthatforblocksof height m +1, j H m +1 j =2+(2 1 ) m r +2 + ::: +(2 c 1 ) m r +2 + c j W 0 c j +( d 1) j W 0 d j .Using 98

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theinductionhypothesis,placeblocksfrom R m intoequivalenceclassesbasedonthe wallpaperpatternthatappearsovertheircolumns.Notetha tallblocksoflength n c havewallpaperpatternsthatappearovertheirrowsaswell. Saywenowwanttouse verticalconcatenationtoaddanewrowtosomeblock B m;n ,producingthenewblock B 0 m +1 ;n .For0 n c 1therearenorestrictionsonthesymbolsthatmayappearin thenewrowoftheblockextension,butfor n c therewillberestrictionsbasedonthe (vertical)wallpaperpattern(unless r =1,whichisinconsequential).Similartotheproof ofLemma5.1.10,weshallconsidercasesbasedonthelength n ofblocksplacedinthe equivalenceclasses. Firstconsideranequivalenceclassrepresentedbyablock B m +1 ;n oflength1 n c 2.Ifweinspecttherightcontextsoftheequivalenceclasse srepresentedbytheseundersized blocks,weagainndthatotherundersizedblocksofvarying lengthsaregroupedinto therightcontextbasedsolelyontheirlengths;thatis,Lem ma1.3.1guaranteesthat ( ";B 0 m +1 ;n 0 ) 2 ([ B m +1 ;n ])whenever n 0 c n 1.Ontheotherhand, B m +1 ;n willagain containblocksoflength n 0 >c n 1initsrightcontext,butnowtheselongerblockswill belongtotherightcontextofanequivalenceclassonlyifth eirwallpaperpatternagrees withthepartialpatternsspeciedby B m +1 ;n as S istranslatedbothhorizontallyand vertically:thatis,for n 0 c ,( ";B 0 m +1 ;n 0 ) 2 ([ B m +1 ;n ])onlyifblock = B m +1 ;n B 0 m +1 ;n 0 hasthepropertythatforall i 2f 0 ; 1 ;:::;n 1 g andall j 2f 0 ; 1 ;:::;m r +1 g P r 1 y =0 ( i;j + y )= P r 1 y =0 ( i + c;j + y ).Forall n< c 2,Corollary1.3.2guarantees that j F 1 ;n ( X ) j =2 n andLemma1.3.1guaranteesthat B m;n n 2 F ( X )forall B m;n 2 F m;n ( X )andall n 2f 0 ; 1 g 1 ;n .Therefore,eachequivalenceclassin H m representedbya block B m;n oflength n c 2creates2 n additionalequivalenceclassesin H m +1 ,giving ((2 n ) m r +1 )2 n =((2 n ) m r +2 )asdesired. Nowconsideranequivalenceclassrepresentedbyablock B m +1 ;n oflength n c 1. Sincehorizontaltranslatesof S mustalwaysfollowadistinctwallpaperpattern, B m +1 ;n will exhibitapatternoflength c eventhoughtheheightisincreasedbyonerow.Therefore wecanapplytheargumentfoundintheproofofLemma5.1.10by adjustingtheonedimensionalalphabettobecomprisedofthesymbols f D;E g m r +1 ; 1 andthenformingthe sets W 0 ;W 0 c ; and W 0 d accordingly. 99

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Thereisaninherentdicultyassociatedwitheortstodete rmine H m precisely.Due tothepresenceofbothhorizontalandverticaltranslateso f S withintherepresentative blocks,atransitionfrom H m to H m +1 doesnotproperlyaccountfortherichnessthat resultsfrominterlacinghorizontaltranslateswithverti calones.Forexample,inadot system X denedbyan r c rectangularshape,considerablockofheight m thatcanbe representedbyanon-primitivewordoftheform w 0 =( w 0 1 w 0 2 w 0 d ) c=d asinProposition 5.1.11.Whenthisblockisextendedtoheight m +1,itneednolongerberepresentedby w 0 =( w 0 1 w 0 2 w 0 d ) c=d ;rather,itismorelikelytonowberepresentedbyaprimitiv eword w 0 c oflength c .Therefore,theprocessofcheckingforprimitivewordsmus tbereadjusted ateachheight. 5.2FollowerSetsandPredecessorSets Totakefulladvantageofthe M F ( X ) constructionandthemachine'sabilitytoreadsymbols inanon-linearfashion,thefollowingdenitionsareputfo rth. Thefollowerset ofastate q 2 G = M F ( X ) isthecollectionoflabelsofblockpaths startingat q .Thatis, F G ( q )= f B m;n : 9 blockpath 0 in G with ( 0 )= B m;n ; 0 (0 ; 0)= q g : Thepredecessorset ofastate q 2 M F ( X ) isthecollectionoflabelsofblockpaths terminatingin q .Thatis, P G ( q )= f B m;n : 9 blockpath 0 in G with ( 0 )= B m;n ; 0 ( n 2 ;m 2)= q g : Weshallsaythatgraph G is followerseparated ifeachstatehasadistinctfollowerset. Foravertexshift G withdistinctlabelsonthestates, G isnaturallyfollowerseparated: inthiscase, q 0 2 F G ( q )ifandonlyif q = q 0 .However,whenthelabelsonthestatesare notdistinct,itmaybethecasethattwodierentstateshave thesamefollowerset.Ifwe considerthe(strictly)socshiftspace Y ofExample2.2.2,aquickinspectionrevealsthat thegraphcorrespondingto Y isfollowerseparatedalthoughthelabelsonthestatesare 100

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notdistinct.Example5.2.1 Recallthestrictlysocshiftspace Y ofExample2.2.2andthefourstates q 1 ;q 2 ;q 3 ,and q 4 ofFigure2.9.Foreaseofreference,weincludeheretherela beledgraph M F ( X ) = G p qq q q 12 3 4 q q q q p Figure5.2:Strictlysocshiftwithfollower-separatedgr aph Nowdenethefollowingfourblocksfrom F ( Y ). 1 = pp qpqp 2 = ppp qqpqqp 3 = pp qp and 4 = ppp qqp Thenfor j 2f 1 ; 2 ; 3 ; 4 g ; j 2 F G ( q i ) i = j .Furthermore,onlystate p canbefollowed bytheblock B 2 ; 2 denedby B 2 ; 2 ( i;j )= p for0 i 1 ; 0 j 1.Therefore,graph G = M F ( X ) isfollowerseparated. Wesaythatstateswiththesamefollowersetare equivalent .Inone-dimensionalsymbolicdynamics,equivalentstatesofagraphcanbemergedto createasmallergraph presentingthesamesocshift.Equivalentstatesin M F ( X ) canalsobemergedwithout aectingtherepresentedshiftspace.Proposition5.2.2 Givenagraph G = M F ( X ) = f Q;E;s;t; g andtheequivalencerelation q q 0 i F G ( q )= F G ( q 0 ) ,partitionthesetofstates Q intodisjointequivalence classes Q 1 ;Q 2 ;:::;Q v .Deneagraph G 0 withstates Q 0 = f Q 1 ;Q 2 ;:::;Q v g andanedge from Q i to Q j exactlywhentherearevertices q 2 Q i and q 0 2 Q j andanedgein G from 101

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q to q 0 .(Thelabelingfunctionandthedenitionofacceptanceare unchanged.)Then L ( G )= L ( G 0 ) Proof. Denoteby Y thetwo-dimensionalshiftspacerepresentedby M F ( X ) .Bydenition, thelanguageoftheoriginalgraph L ( G )= L ( M F ( X ) )= F ( Y )istheunionofallfollower setsofstatesin G .(Allblocksinthelanguageofthegraphstartwith = q forsome state q 2 Q .)Thelanguageofallblocksrecognizedby G 0 isalsotheunionofthefollower setsofstatesin G 0 :denotethisby L ( G 0 )= F ( Y 0 ).Sincethefollowersetsarethesame, thesetofallrecognizedblocksarethesamesothat F ( Y )= F ( Y 0 ).Finally,sincethe languageofashiftspaceuniquelydeterminestheshiftspac e,itmustbethecasethat Y = Y 0 Themergedgraph G 0 neednotconformtotheforbiddenandforcedstructuresoutl ined inSection2 : 3:Inparticular,itispossiblethatgraphdiamondswillbec reatedduringthe mergingprocess.Thisisonlypossiblebecausethemergedgr aphofastrictlysocshift space Y willhavefewerstatesthantheoriginalgraph M F ( X ) representingthepreimage X of Y .Forthisreason,thetechniqueofmergingstatesasinPropo sition5.2.2must bedistinguishedfromthatofstateamalgamtionusedinChap ter4.Stateamalgamation wasappliedtomergestateswiththesamelabelinagraphderi vedfromtheblockpath representingapointinashiftofnitetype.Inthatsetting ,stateamalgamationwasused toreduceagraphofsize ab representingaperiodicpointofdoubleperiod( a;b )inorder todeterminethesizeofthesubgraphcontainedwithin M F ( X ) ,wherestatesareknownto havedistinctlabels.Inthepresentsetting,statesaremer gedinagraphrepresentinga socshiftspaceinaneorttondagraphofsmallersizethat iscapableofrecognizing thesameshiftspace.Notethatasconstructed, M F ( X ) recognizingatwo-dimensionalshift ofnitetype X cannotbefurtherreducedsinceeachstateisadistinctelem entintheset ofblocksthatdescribethespace.Itisonlypossibletoredu cethesizeofthegraph M F ( X ) representinga(strictly)socshiftthatistheimageof X underablockcodesinceseveral statesmaynowhavethesamelabel.However,statesinagraph representingasocshift thathavethesamelabelcanbemergedonlyifithasbeendeter minedthatthesestates havethesamefollowerset.Ingeneral,thiscanbedicultto determine.Thefollowingis asimpleexampleintheone-dimensionalcasethatcanbegene ralizedtoatwo-dimensional 102

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settinginordertoillustratethemergingofstates.Example5.2.3 Let X betheone-dimensionalstrictlysocshiftspacecomprised ofthe bi-innitesequencesthatcontainnomorethanone1.(If X wereashiftofnitetype, therewouldbeanitesetofblocks B X suchthatabi-innitesequence x wouldbeapoint in X ifandonlyifallsubblocksof x belongto B X .Foranitesetofblocks,therewould beamaximumlength N forblocksbelongingtotheset B X .Thenthebi-innitesequence 0 10 N 10 = 2 X althoughallsubblocksof x wouldbelongto B X .Sothereisnonitesetof blocksdescribing X sothat X mustbestrictlysoc.)Agraphrepresenting X vialabeled statesisprovidedinFigure5.3.Thereisneithervertexshi ftnoredgeshiftrepresenting X ,sincebothvertexshiftsandedgeshiftsareessentiallysh iftsofnitetype[25].Notice, however,thatthestatesofthegrapharefollowerseparated 10 0 Figure5.3:Graphofone-dimensionalstrictlysocshift Example5.2.3canbegeneralizedtotwodimensionstocreate agraphwithstateswhose followersetsareeasytodetermine.Example5.2.4 Considertheshiftofnitetype X presentedbythevertexshift M F ( X ) ofFigure5.4.Nowdenethe1 1blockcode by ( z )= 8<: 1:if z =1 0:otherwise Thenthesocshiftspace Y isthesetofallcongurationsoftheplanecontainingatmos t one1.Infact, Y isstrictlysoc.Toseethis,supposetowardsacontradicti onthat Y isa shiftofnitetype.Thentheremustexistsome N 1suchthat Y := f y 2f 0 ; 1 g Z 2 : 8 v 2 Z 2 ; v ( y ) [ N;N ] 2 Q = F N;N ( Y ) g : (5.2.1) Nowconsiderthepoint y ( i;j )= 8<: 1for( i;j ) 2f (0 ; 0) ; ( N +2 ;N +2) g 0otherwise : (5.2.2) 103

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ddd d dBd B BeB e eee e ddD D dBD 1 Be1 E eeE E DDa a D1a A 1EA b EEb b aaa a aAa A AbA b bbbb q 12 q 14 q 13 q 15 q 8 q 9 q 10 q 11 q 4 q 5 q 6 q 7 q 0 q 1 q 2 q 3 Figure5.4:Strictlysocshift;graphnotfollowerseparat ed Then y issuchthat 8 v 2 Z 2 ; v ( y ) [ N;N ] 2 Q = F N;N ( Y ),but y= 2 Y sincethiscongurationoftheplanecontainsmorethanone1. Thegraph M F ( X ) = G isnotfollowerseparated.Toseethis,let y 0 betheconguration oftheplanepopulatedentirelywith0'sandlet F ( y 0 )denoteallfactorsofthispoint, i.e.,thesetofallblockspopulatedentirelywith0's.Then for i 2f 3 ; 7 ; 11 ; 12 ; 13 ; 14 ; 15 g F G ( q i )= f F ( y 0 ) g .Wecanmergethesestatesingraph G tocreateasmallergraph G 0 representingthesameshiftspace Y .Noticethatseveralstates q i havethesamepredecessor setaswell:if i 2f 0 ; 1 ; 2 ; 3 ; 4 ; 8 ; 12 g ,then P G ( q i )= f F ( y 0 ) g .Alternativelythen,wecould havemergedthesestatesingraph G tocreateasmallergraphrepresentingthesameshift space Y .However,ifweattempttomergethesestatesnowinthegraph G 0 ,thenewgraph willnolongerrepresenttheshiftspace Y aspointswillbeacceptedthatcontainmore thanasingle1.Noticethatonlystates q 3 and q 12 havebothpredecessorandfollowerset equalto F ( y 0 ).Soifwemergestates q i for i 2f 0 ; 1 ; 2 ; 4 ; 8 g tocreategraph G 00 fromgraph G 0 ,wewillhavereducedthesizeofthegraphwhilestillmainta iningdistinctinitialand terminalstatesforblocksinthelanguage F ( Y )thatcontainasingle1.Inotherwords,the graph G 00 represents Y .SeeFigure5 : 5forthemergedgraphwithstateslabeledaccording to 104

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000 0 000 1 001 0 010 0 100 0 0000 Figure5.5:Graphofreducedsizerecognizing Y Withthisintroductiontothetopicoffollowersets,wenowh aveallthetoolsnecessary todiscussthepropertiesofgraphsrepresentingtwo-dimen sionalsocshiftspacesandtheir correspondingfactorlanguages.Thegraphrecognizing Y inFigure5 : 5isnottransitivenoristhelanguage F ( Y ),sinceablockcontainingthesymbol1cannotmeetitselfin any direction.Thegraphisalsonotdeterministicsince(forex ample)thebottom-leftstatehas twohorizontaltransitionswiththesamelabel.Althoughth egraphisfollowerseparated, thebottom-leftandtop-rightstatesbothcontainblocksof arbitrarysizepopulatedwithall zerosintheirfollowersets.Thismeansthatthereisnotaon e-to-onerelationshipbetween grid-innitepathsofthegraphandpointsoftheshiftspace ,asevidencedbymultiple grid-innitepathsrepresentingthesameperiodicpointof allzeros.Example5.2.4has broughtusfull-circleinoneothersense:blocksinthelang uage F ( Y )maybesurrounded byaborderofuniformsymbols(inthiscase,thesymbol0)ina fashionreminiscentofthe #symbolusedtosurroundpicturesinREC.Thestudyofrecogn izablepicturelanguages hasnowbeensuccessfullyplacedwithinthecontextoftwo-d imensionalsocshiftspaces. 105

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Conclusion Intheliterature,investigationssurroundingtheclassRE Cofrecognizablepicturelanguageshavebeencarriedoutintermsoflanguagetheory.Thi smotivatedthecurrent researchwithrespecttothesetofallfactors(blocks)ofat wo-dimensionalsocshift space.Inone-dimensionallanguagetheory,thereareagrea tmanyresultsregardingFTR (factorial-transitive-recognizable)languages,sothat theearlyfocusofthisresearchwas ontransitivity.Itquicklybecameapparentthattherewasa needtoclarifywhatwas meantbytransitivityinthetwo-dimensionalcase.Thedie rentnotionsoftransitivityin theclassofrecognizablelanguageshavenotbeenstudiedbe fore.Evenfordotsystems thereareseveralquestionsabouttransitivitythatremain unanswered,asmanyshapes stilleludeclassicationatthistime.Dotsystemshavepro videdexamplesoflanguages thathavedirectionaltransitivitybutnotuniformlyso,an dlanguagesthataremixingbut notuniformlyso.Missingisanexampleofalanguagethatist ransitivebutnotuniformly so.(IthankAntonioRestivoforposingthequestionatthe20 06conferenceonAdvances onTwo-dimensionalLanguageTheory.)Theclassicationof two-dimensionallanguages basedonaheirarchyoftransitivitypromisestobeanintere stingexperiment;forexample, Proposition3.1.8providesalanguagethatisuniformlytra nsitiveyetfailstobetransitive incertaindirectionsandthereforecannotbemixing.Thech aracterizationof(uniformly) transitivedotsystemsand,ingeneral,localandrecogniza blelanguagesremainsopen. Thegraphthatisintroducedwiththispaperisthersttosuc cessfullyrepresent shiftsofnitetypeaswellastheirsocimages.Recognitio nofthepointsina(strictly) socshiftspacerequiresthatthescanningmechanismbeabl etoalternatethereading ofsymbolsinthehorizontalandverticaldirections.Thism akesthegraphsconstructed hereinanimportanttoolforthestudyoftransitivityinthe representedfactorlanguages andperiodicityintherepresentedshiftspaces.Forexampl e,thegraphprovidesanice 106

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mechanismtoproveTheorem4.1.4relatinguniformhorizont altransitivitytoperiodicity. Becauseoftheemptinessproblem,questionsconcerningper iodicityarecentraltothe discussionoftwo-dimensionalsocshiftspaces.Furtherw orkneedstobedonewith respecttoperiodicpointsandtheirappearanceingraphsre presentingtwo-dimensional shiftspaces.Itremainstobeseenwhatotherpropertiesoft heshiftspacecanandcannot beobservedfromthegraphpresentationconstructedhere.( However,anypropertythat canbeobservedfromthegraphstructurewillnecessarilybe decidable.)Inparticular, itremainsunclearwhatthenecessaryandsucientconditio nsareforatwo-dimensional graphrepresentationtobearepresentationofatransitive picturelanguage. Theuseofasyntacticmonoidofequivalenceclassesforbloc ksinthefactorlanguage hi-litestherelevanceoftransitivitytothepropertiesof atwo-dimensionallanguage.It alsoexposestheshortcomingsoftreatingtwo-dimensional languagesasone-dimensional languagesbylimitingtheheightofblocksunderconsiderat ion.Thisavenueofresearch nowseemsperipheraltothestudyofrecognizablelanguages denedbytwo-dimensional shiftspaces,althoughthetopicbearssomeinterestinitso wnright. Theuseofequivalenceclassesforstatesinagraphcanleadt othemergingofstates, whichmakesthegraphsizemoremanageableandalsoopensthe doorforquestionsregardingtheminimaldeterministicgraphrepresentationof ashiftspace.Amainresultin one-dimensionalsymbolicdynamicsisthatforasocshifts pace X withfactorlanguage thatistransitive,adeterministicgraphistheminimaldet erministicpresentationof X if andonlyifthegraphistransitiveandthestatesofthegraph havedistinctfollowersets [25].Itshouldbedeterminedwhetherasimilarstatementap pliesfortwo-dimensional socshiftspaces.TheshiftspaceofExample5.2.4,whileno ttransitive,isastepin therightdirectiontowardlinkingtheseconceptsinthetwo -dimensionalcase.Moregenerallanguagesresultingfromblockcodesofthesetoflangu agesthatfullltheproperty A ( X )= F ( X )needtobestudiedinregardstothetopicofmergingstates. Two-dimensionalsymbolicdynamicsisvariedenoughfromth eone-dimensionalcaseto warrantacompletetreatment(e.g.textbook)alongtheline softheoutstandingresource, AnIntroductiontoSymbolicDynamicsandCoding byLindandMarcus[25].Theworkin thisdissertationprovidesasolidfoundationtoconnectth ekeyelementsoflanguagetheory, automatatheory,andsymbolicdynamicsastheyrelatetothe classoftwo-dimensional factorialandprolongablerecognizablelanguages. 107

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AbouttheAuthor JoniBurnettePirnotreceivedherBachelorofArtsinMathem aticsfromNewCollegeof FloridaasaNewCollegeFoundationScholarandaFloridaAca demicScholar.In1997, shereceivedherMasterofArtsinMathematicsfromtheUnive rsityofSouthFloridaand beganteachingatManateeCommunityCollege(MCC).Shecurr entlylivesinSarasota, Florida,withherhusbandandtwoteenagedaughtersandisan AssistantProfessorof MathematicsatMCC.


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Recognizable languages defined by two-dimensional shift spaces
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ABSTRACT: There are numerous connections between the theory of formal languages and that of symbolic dynamics. In each, the transition from one dimension to two dimensionsis accompanied by much difficulty due in large part to the emptiness problem, which is related to the presence (or lack thereof) of periodic points and is known to be undecidable. Here, we focus on two-dimensional languages that have the property that all blocks allowed by the language can be extended to a configuration of the plane satisfying the structure of the language; for such languages the emptiness problem is not an issue. We first show that dot systems may be associated with two-dimensional languages having this property, so that we might employ these languages as varied examples. We next define a new type of finite automaton and with it, a tool for recognizing two-dimensional "strings" of data. It is then shown that these automata correctly represent the sofic shift spaces that result from the application of block maps to shifts of finite type. Thereafter, these automataare utilized to investigate properties of transitivity in the two-dimensional languages that they represent. More specifically, new definitions for different types of two-dimensional transitivity are adapted from topological dynamics and then illustrated through the use of dot systems. The appearance of periodic points in the languages represented by these automata is also explored, with a main result being that the existence of a periodic pointis guaranteed under certain conditions. Finally, issues of equivalence are introduced in the two-dimensional setting with regards to formal languages (syntactic monoids) and symbolic dynamics (the follower sets of a graph representing a sofic shift space).
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Recognizable picture languages.
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Transitivity.
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