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Chaotic Computation

Permanent Link: http://ufdc.ufl.edu/UFE0024234/00001

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Title: Chaotic Computation
Physical Description: 1 online resource (147 p.)
Language: english
Creator: Miliotis, Abraham
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: cantor, chaos, computation, logic, search
Biomedical Engineering -- Dissertations, Academic -- UF
Genre: Biomedical Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: Chaotic Computation is the exploitation of chaotic systems to perform computational tasks. The abundance of uncountable distinct behaviors by chaotic systems, along with their embedded determinism, position such systems as perfect candidates for developing a new computational environment. The present dissertation focuses on algorithms developed over the past decade within the realm of Chaotic Computation. After a brief exposition of general Chaos Theory, we proceed to give detailed instructions for performing such algorithms, as well as specific examples of implementations. We begin with multiple methods for number representation and basic arithmetic manipulations, providing from the start evidence of the flexibility of Chaotic Computation. The compatibility with Turing machines is subsequently shown through an algorithm for logic operations whose general form is a recurrent theme. We soon, though, proceed further than Turing machines and present a solution to the Deutsch-Jozsa problem, of arbitrary binary functions. Even more, a practical issue is also handled by showing how chaotic systems have a natural way for selecting matches of a searched item from within an unsorted database. Finally we present our latest results in handling 'prolonged' evolution of chaotic systems. Specifically we demonstrate the dominance of selecting the appropriate behavior, for a computational task, over being exact with specific state values, or even confined to specific physical quantities.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Abraham Miliotis.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Ditto, William L.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024234:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024234/00001

Material Information

Title: Chaotic Computation
Physical Description: 1 online resource (147 p.)
Language: english
Creator: Miliotis, Abraham
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: cantor, chaos, computation, logic, search
Biomedical Engineering -- Dissertations, Academic -- UF
Genre: Biomedical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Chaotic Computation is the exploitation of chaotic systems to perform computational tasks. The abundance of uncountable distinct behaviors by chaotic systems, along with their embedded determinism, position such systems as perfect candidates for developing a new computational environment. The present dissertation focuses on algorithms developed over the past decade within the realm of Chaotic Computation. After a brief exposition of general Chaos Theory, we proceed to give detailed instructions for performing such algorithms, as well as specific examples of implementations. We begin with multiple methods for number representation and basic arithmetic manipulations, providing from the start evidence of the flexibility of Chaotic Computation. The compatibility with Turing machines is subsequently shown through an algorithm for logic operations whose general form is a recurrent theme. We soon, though, proceed further than Turing machines and present a solution to the Deutsch-Jozsa problem, of arbitrary binary functions. Even more, a practical issue is also handled by showing how chaotic systems have a natural way for selecting matches of a searched item from within an unsorted database. Finally we present our latest results in handling 'prolonged' evolution of chaotic systems. Specifically we demonstrate the dominance of selecting the appropriate behavior, for a computational task, over being exact with specific state values, or even confined to specific physical quantities.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Abraham Miliotis.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Ditto, William L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024234:00001


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CHAOTICCOMPUTATION By ABRAHAMMILIOTIS ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c 2009AbrahamMiliotis 2

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TABLEOFCONTENTS page LISTOFTABLES .....................................5 LISTOFFIGURES ....................................7 ABSTRACT ........................................10 CHAPTER 1INTRODUCTION ..................................11 1.1DissertationOverview .............................14 1.2Chaos ......................................15 1.2.1LogisticMap,TopologicalTransitivityandPeriodThree ......15 1.2.2TheTentMap,TopologicalConjugacyandUniversality .......38 1.2.3ThresholdControlandExcessOverowPropagation .........43 1.3Conclusion ....................................47 2INTRODUCTIONTOCHAOTICCOMPUTATION ...............49 2.1NumberEncoding ................................49 2.1.1ExcessOverowasaNumber .....................49 2.1.2PeriodicOrbitsforNumberRepresentation ..............50 2.1.3RepresentationofNumbersinBinary .................52 2.2ArithmeticOperations .............................53 2.2.1DecimalAddition ............................53 2.2.2BinaryAddition .............................56 2.2.3DecimalMultiplicationandLeastCommonMultiple .........59 2.3BinaryOperations ...............................60 2.3.1LogicGates ...............................61 2.3.2ParallelLogicandtheHalfAdder ...................64 2.3.3TheDeutsch-JozsaProblem ......................67 2.4Conclusion ....................................73 3SEARCHINGANUNSORTEDDATABASE ....................76 3.1EncodingandStoringInformation ......................77 3.2SearchingforInformation ...........................80 3.3Encoding,StoringandSearching:AnExample ................83 3.4Discussion ....................................84 4ASIMPLEELECTRONICIMPLEMENTATIONOFCHAOTICCOMPUTATION 90 4.1AnIteratedNonlinearMap ..........................90 4.2ThresholdControlChaosintoDierentPeriods ...............90 4.3ElectronicAnalogCircuit:ExperimentalResults ...............92 3

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4.4FundamentalLogicGateswithaChaoticCircuit ..............97 4.5EncodingandSearchingaDatabaseUsingChaoticElements ........100 4.6Conclusion ....................................104 5LOGICOPERATIONSFROMEVOLUTIONOFDYNAMICALSYSTEMS ..106 5.1GenerationofaSequenceof(2-input)LogicGateOperations ........106 5.2TheFullAdderand3-InputXORandNXOR ................110 5.3Conclusion ....................................113 6MANIPULATINGTIMEFORCOMPUTATION .................115 6.1Introduction ...................................115 6.1.1FlexibleLogicGates ..........................115 6.1.2SearchAlgorithm ............................120 6.2NeuralImplementation ............................122 6.2.1NeuralModels ..............................122 6.2.2AlgorithmImplementations .......................127 6.3ElectronicImplementation ...........................132 6.4Discussion ....................................136 7CONCLUSION ....................................139 REFERENCES .......................................141 BIOGRAPHICALSKETCH ................................147 4

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LISTOFTABLES T able page 1-1Summaryofthetransitionsinbehaviourofthedierentintervalsdescribedin Section1.2 .......................................37 1-2Topologicalconjugacy. ................................41 1-3ExperimentalmeasurementsofFeigenbaum'sconstant( )indierentsystems basedontheirperioddoubling. ...........................44 2-1Truth-tableforAND,OR,XOR,NOR,NAND,NOT,andWIRE. ........63 2-2NecessaryandsucientconditionsforachaoticelementtosatisfyAND,OR, XOR,NOR,NAND,NOT,WIRE. .........................63 2-3Initialvalues, x prog ,andthresholdvalues, x ,requiredtoimplementthelogic gatesAND,OR,XOR,NOR,NAND,NOT,andtheidentityoperation(WIRE), with =0 : 25. .....................................64 2-4TruthtableforXORandANDlogicgatesonthesamesetofinputs. ......65 2-5TruthtablefortwoANDgatesoperatingonindependentinputs. ........66 2-6RequiredconditionstosatisedparallelimplementationoftheXORandAND gate. ..........................................66 2-7RequiredconditionsforimplementingtwoANDgatesonindependentsetsof inputs. .........................................66 2-8Examplesofinitialvalues, x prog ;y prog ,andthresholds x ;y ,yieldingtheparallel operationofXORandANDgates. .........................67 2-9Examplesofinitialvalues x prog ;y prog ,andthresholds,yieldingoperationoftwo ANDgatesonindependentinputs. .........................67 4-1Truth-tableforthevefundamentallogicgatesNOR,NAND,AND,ORand XOR. .........................................98 4-2Necessaryandsucientconditionstobesatisedbyachaoticelementinorder toimplementthelogicaloperationsNOR,NAND,AND,ORandXOR. .....99 4-3Numericalvaluesof x prog forimplementinglogicaloperationsNOR,NAND,AND, ORandXOR. .....................................99 4-4Updatedstatevalues, x 1 = f ( x 0 ),ofachaoticelementinordertoimplement thelogicaloperationsNOR,NAND,AND,ORandXOR. ............99 5-1ThetruthtableofthevebasiclogicoperationsNAND,AND,NOR,XOR,OR. 107 5

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5-2Necessaryandsucientconditionstobesatisedbyachaoticelementinorder toimplementNAND,AND,NOR,XORandORonsubsequentiterations. ...108 5-3Thetruthtableoffulladder,andnecessaryconditionstobesatised. ......112 5-4Thetruthtableofthe3-inputXORandNXORlogicoperations,necessaryand sucientconditionstobesatisedbythemap. ..................114 6-1Thetruthtableofeachofthevefundamentallogicgates,AND,NAND,OR, NOR,XOR ......................................118 6-2Timevalues,inarbitrarytime\units"foreachofthevegatesconsidered ...119 6-3Appropriatetimesampleinstances,basedonsimulationpoints,toperformeach ofthevegatesconsidered ..............................128 6-4Appropriate R valuessotimeshiftanactionpotentialinordertoperformeach ofthevegatesconsidered ..............................131 6-5DelaytimesforVstoimplementeachofthevegatesconsidered ........134 6

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LISTOFFIGURES Figure page 1-1BifurcationdiagramfortheLogisticmap. .....................18 1-2ForwardandBackwardevolutionof F 4 .......................19 1-3Indicativebehaviourof I undermultipleapplicationsof F for,(a) < 1and (b)1 < .......................................22 1-4Plotsof F 2 and F 2 : 5 .................................25 1-5Exhibitionofthe\trapping"oftwopointsintwodierentcongurationsof F 28 1-6Renormalizationoftwocasesof F 2 .........................29 1-7Demonstrationofbirthofoddperiodicityxedpoints. ..............33 1-8Plotofthefunction F : I I ............................36 1-9Plotofthefunction T : I I ............................39 1-10BifurcationdiagramfortheTentmap. .......................40 1-11Topologicalconjugacybetweenevolvedstates,upto n =5. ............42 1-12Logisticmapbifurcationdiagramforsomevalueswithin3 << 1 ......44 1-13ThresholdControlMechanism. ...........................46 1-14Thresholdvaluesforconningthelogisticmaponorbitsofperiodicity2to50. .47 2-1Emittedexcessbythresholdingthelogisticmapintheinterval[0 ; 0 : 75]. .....51 2-2Encodingthesetofintegers f 0 ; 1 ;:::; 100 g .....................52 2-3Numberencodinginbinaryformat. .........................54 2-4SerialAddition. ....................................55 2-5Decimalparalleladdition. ..............................57 2-6Thebranchingalgorithmcanbeextendedtoalargertreelikestructure. .....57 2-7Schematicrepresentationoftheserialadditionmethodforbinarynumbers. ...58 2-8Schematicrepresentationoftheparalleladditionmethodforbinarynumbers. ..59 2-9SchematicrepresentationofthemethodfortheLeastCommonMultipleoffour numbers. ........................................61 2-10Basisfunction T -TentMap. ............................71 7

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2-11Basisfunction e T -InvertedTentMap. .......................72 2-12FourrealizationsofthechaoticDeutsch-Jozsaalgorithmforthecase k =3. ...74 2-13Thetotalexcessemittedfromeachofthe72functions. ..............75 3-1TheTentmapunderthethresholdmechanism. ..................78 3-2Schematicrepresentationofthechangesinthestateofdierentelements. ....82 3-3Searchingfor\ l ". ...................................85 3-4Searchingfor\ e ". ...................................86 3-5Searchingfor\ x ". ...................................87 4-1Bifurcationdiagramoftheiteratedmapforvariousvaluesof and .....91 4-2Graphicalformofthemaptobeimplementedbyanelectroniccircuit ......92 4-3Eectofthresholdvalue x onthedynamicsofthesystem. ............93 4-4CircuitdiagramofthenonlineardeviceofEquation4{3 ..............94 4-5Voltageresponsecharacteristicsofthenonlineardevice. ..............95 4-6Schematicdiagramforimplementingthethresholdcontrollednonlinearmap. ..96 4-7Circuitdiagramofthethresholdcontroller. .....................96 4-8PSPICEsimulationresultsoftheexperimentalcircuit. ..............97 4-9Searchingfor\ b ". ...................................103 4-10Searchingfor\ o ". ...................................104 4-11Searchingfor\ d ". ...................................105 5-1GraphicalrepresentationofveiterationsoftheLogisticmap. ..........109 5-2Patternsofbinarytwoinputsymmetricoperations. ................111 6-1Schematicrepresentationofaexible2-inputlogicgate ..............116 6-2Constructionofa\generic"signal ..........................119 6-3Schematicrepresentationofthetimebasedsearchmethod ............122 6-4TimeDelayUnit(TDU) ...............................126 6-5DemonstrationofoperatingaNORgate,withasingleneuronusingdierent samplingtimes ....................................128 8

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6-6DemonstrationofoperatingaNORgate,withaneuralcircuitusingdierent delaytimes ......................................130 6-7Schematicrepresentationofanelectroniccircuitforlogicusingtime .......133 6-8DemonstrationofoperatingaNORgateusinganelectroniccircuitutilizingtime dependantcomputation ...............................134 6-9Schematicrepresentationofanelectroniccircuitforthetimedependantsearch method ........................................136 6-10Demonstrationofperformingasearchusinganelectroniccircuitutilizingtime dependantcomputation ...............................137 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy CHAOTICCOMPUTATION By AbrahamMiliotis May2009 Chair:WilliamL.Ditto Major:BiomedicalEngineering ChaoticComputationistheexploitationofchaoticsystemstoperformcomputational tasks.Theabundanceofuncountabledistinctbehavioursbychaoticsystems,alongwith theirembeddeddeterminism,positionsuchsystemsasperfectcandidatesfordevelopinga newcomputationalenvironment.Thepresentdissertationfocusesonalgorithmsdeveloped overthepastdecadewithintherealmofChaoticComputation.Afterabriefexposition ofgeneralChaosTheory,weproceedtogivedetailedinstructionsforperformingsuch algorithms,aswellasspecicexamplesofimplementations. Webeginwithmultiplemethodsfornumberrepresentationandbasicarithmetic manipulations,providingfromthestartevidenceoftheexibilityofChaoticComputation. ThecompatibilitywithTuringmachinesissubsequentlyshownthroughanalgorithm forlogicoperationswhosegeneralformisarecurrenttheme.Wesoonthough,proceed furtherthanTuringmachinesandpresentasolutiontotheDeutsch-Jozsaproblem,of arbitrarybinaryfunctions.Evenmore,apracticalissueisalsohandledbyshowinghow chaoticsystemshaveanaturalwayforselectingmatchesofasearcheditemfromwithin anunsorteddatabase. Finallywepresentourlatestresultsinhandling\prolonged"evolutionofchaotic systems.Specicallywedemonstratethedominanceofselectingtheappropriate behaviour,foracomputationaltask,overbeingexactwithspecicstatevalues,or evenconnedtospecicphysicalquantities. 10

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CHAPTER1 INTRODUCTION Overthelastvedecadestheadvancementofcomputationalmachineshasbeen aninvaluableachievementbyoursocietyandextremelybenecial.Moore`slaw[ 1 ] eitherasaself-fulllingprophecy,orasasimpledescriptionofthisprogress,hasbeen obeyeduptonowclosely.Theimmenseimportanceofpreservingthisprogresshasdriven manytoinvestigatepossiblereasonsthatcouldhinderfurtherincreaseincomputational performance. Obviouslytheultimatelimitsimposedbythephysicalnatureofourworldwere thersttobeidentied.Thethreemostimportantincludethespeedoflightlimiton transmissionofinformation,thelimitontheamountofinformationanitesystem canstore,andthethermodynamiclimitofthermalenergydissipationbytheerasureof information[ 2 ],thelatterbeingthemostrelevanttocurrentsemiconductortechnology. Perhapsmostimportantthough,atthistime,arethelimitationsinherentin semiconductortechnology,whichisthebasisofcontemporarycomputers.Regularly overthelastdecadetheInternationalTechnologyRoadmapforSemiconductors(ITRS) hasbeenpublishedwiththeaimtoindicatethetreadofspecicfeaturesofsemiconductor technology[ 3 ],forexamplelithographyresolution,transistorsizeandconnectingwire diameter.Currenttechnologyis\relatively"farfromfundamentallimitsforthese attributes(molecularsizesandthermalnoise),butapproachingthematanalarming rate;forexampleAustinet.al.reportamethodforcreatingmemoryhalf-pitchsize equivalenttothesizeofaninsulinmolecule(6nm)[ 4 ]. Astheneedforalternativecomputationalparadigmsisbecomingprogressively evidentmanyphysicalsystemshavebeenproposedasalternativebasisforcomputational machines.Themostprominentbeingquantumcomputing[ 5 6 ]andDNAcomputing [ 7 { 10 ].Evenaftermanyyearsofintenseresearchthough,bothquantumandDNA computingarestillfacingsomefundamentalproblems.Themainproblemwithboth 11

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thoseparadigmsisthattheyareconned,bytheirnature,toasinglephysicalrealization. Ineachcasethisconnementpresentsdierentproblemsthatarediculttoovercome. Forquantumcomputingthedominantissueisthatofthermalnoise,whichrandomizes individualqubitsandleadsto\decoherence"ofthesystem.DNAcomputingismainly restrictedbythetime-scalewithwhichthenecessarychemicalreactionstakeplace, makinganycomputationaltaskveryslowinproducingaresult. Regardlessofthephysicalsystemusedforcomputationitisclearthattherewillbe ultimatelimitationsineectivenessforaccomplishinganyindividualtask.Thecommon themeinallalternativecomputationalparadigmsisexibilityandparallelization;this isalsopursuedbyconventionalcomputerscienceresearch.Thecommongoalisthe constructionofamethodologythatwillprovideacomputationalmodelthatwillbeable tosolveeachcomputationalproblemthemostecientwayand,ifpossible,multiple problemssimultaneously.Themaindrivingforcetoachievesuchamodelisthefactthat themostpowerfulcomputationalmachinewehave,thehumanbrain,isabletoboth,solve problemsinmultipleways,andhandlemultipleproblemssimultaneously. Chaoticsystemsoverthelastfewdecadeshaveattractedtheattentionofalarge portionofthescienticcommunity.Themainreasonistheabundanceofsuchsystemsin natureandtheextensiverepertoireofbehaviourstheyexhibit[ 11 12 ];compoundedbythe factthattheyaredeterministicsystemsandcanbedescribedbyasmallsetofequations. Morerecentlytherehasbeenintenseresearchinmethodstocontrolchaoticsystems, inspiredprimarilybytheworkofOtt,GrebogiandYorke[ 13 ].Controlmechanismshave providedthemeanstoconneachaoticsystemtoaspecicbehaviour.Suchtechniques havebeentakenadvantagebymanyeldsofcontemporaryresearchincludingcontrol, synchronization,communicationsandinformationencoding[ 14 15 ]. Manyattemptshavebeenmadesincebeforethe90`s,andmanyarestillactive eldsofresearch,forbridgingdynamicsandcomputation[ 16 { 23 ].Probablythough theclearestsuggestionthatachaoticsystem,specically,canbeusedforcomputation 12

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canbeattributedtoC.Moore[ 24 ].Itwasn'tuntillate90`sthough,thattheuseofa non-feedbackcontrolmechanismonachaoticsystemtoperformspeciccomputational taskswasdemonstrated[ 25 26 ].Overthelastnineyearsnon-feedbackcontrolhasbeen utilizedtoshowhowchaoticsystemscanperformarithmeticandbinarylogicoperations, andevenfurthertosolvemorecomplexproblemsliketheDeutsch-Jozsaproblemand searchinganunsorteddatabase[ 27 28 ].Non-feedbackcontroltypicallyisachievedeither bytheuseofathresholdonastatevariableorbyselectingspecicvaluesforthesystem parameters,ineithercaseconningasystemtoaspecicsubsetofallavailablepoints. Thekeyideaisthatfromthewidevarietyofbehavioursembeddedinachaotic systemonecanndaspecicpatternofbehaviourthatcanaccomplishaspecictask. Controlcanconneasystemtotherequiredspecicbehaviouralpattern,withoutloss oftheabilitytoswitchtoadierentbehaviourandperformadierenttask.Thisis accomplishedbyrigorousinvestigationofachaoticsystemtoidentifytherequired behavioursthatcanreliablyrepresentcomputationaltasks;andthestatevariable thresholds,orparametervalues,thatwillconnethesystemtoevolveintherequired manner.Thisallowsustoenvisionacomputationalmachinewhichhasasitsbuilding blockachaoticsystem;eachelementbeingidenticaltoallothers,providingredundancy andreliability;eachelementabletoperformamultitudeoffunctions,providingexibility; andeachelementindependentofallothers,providingparallelismoftasks. ThereasonwearespecicaboutwhichchaoticsystemtouseistheUniversalityof chaos[ 29 ].Chaoticsystems,eveniftheirgoverningequationsdier,behaviourallyare thesame.Thisnotonlyallowsustochoosethemostconvenientsystemfortheoretical developmentofalgorithms,butmoreimportantlydoesnotconnethephysicalrealization toaspecictechnology.Asmentionedabove,chaoticsystemsexistinabundanceinnature includinghigh-speedelectroniccircuits,lasers,andevenneurons;anyofwhichcanbeused asthebuildingblockofachaosbasedcomputer. 13

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1.1DissertationOverview Thisdissertationcanbeconsideredtobeoffourparts.Therstpartistheremaining ofthisintroductorychapter,thesecondpartchapterstwoandthree,whilethefourth chaptercanbeconsideredtobeonitsown.Finallychaptersveandsixcomprisethe nalpartofourexpositionofChaoticComputation. IntheremainingofthisintroductorychapterweprovidenecessarypartsofChaos Theorysothattorelateitwithcomputation.Specicallyweexposethereadertoaview ofChaosTheoryfroma\loose"SetTheoryapproach,withtheintensiontoimplytothe readertheconnectionsfromSetTheory,toChaosTheory,toComputationandgeneral MathematicalLogic.AsthisdissertationisconcernedsolelywithChaoticComputation wedonotdelveindepthontheperipheralissuesanddirectthereadertoappropriate references[ 30 31 ]. Chapterstwoandthreecanbeconsideredtobethemainpartofthisdissertation; theyareanexpositionofestablishedalgorithmsofChaoticComputation[ 25 { 28 32 33 ]. WebegininchaptertwowiththeearliestalgorithmsofChaoticComputationdeveloped overthelastnineyearsbyDittoet.al.,algorithmsfornumberrepresentation,arithmetic operationsandasolutiontotheDeutsch-Jozsaproblem.Whilethethirdchapterpresents exclusivelytherecentalgorithmforsearchinganunsorteddatabase[ 28 ]. Intherelativelysmallthirdpartthatischapterfour,wepresentarecentimplementation ofmostabilitiesofChaoticComputationwithanextremelysimpleelectroniccircuit[ 34 ]. Despiteitslength,theaimsofthispartaretwofold;rstandforemosttogivesubstance ofrealizationtothewholeconceptofChaoticComputationthroughaphysicalsystem implementation;andbeyond,topresentaconcretedemonstrationoftheUniversalityof chaos,andtheeasewithwhichwecantranslateourresultstoanychaoticsystem. Thenalpartisourmostrecentresults,the\cuttingedge"developmentsinChaotic Computation[ 33 ].Chapterve,whilefocusessolelyonbinaryoperations,introduces theimportanceofwhatcanbeconsideredtobe\thetimedimension";inadditionit 14

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expandsonourtreatmentof\statevalues",byshowingmoreconcretelytheimportance ofspecicbehavioursoverexactstatevalues.Finallychaptersixexpandsfurtherthe ideaofmanipulatingtime.Throughdemonstrations,withelectroniccircuits,andmainly neuralcircuitsweexhibitimplementationsofChaoticComputationalgorithmsusing\time instances"asthemediumforcomputationalcommands. Wewishtothereadertheencounterwiththisexpositiontobebothinformativeand enjoyable. 1.2Chaos ChaosTheoryisthethirdmajor\revolution"ofPhysicsofthe20 th Century.Even thoughits\birthdate"canbeplacedascontemporarytoRelativityandQuantum Mechanics,circa1900,itwasn'tuntilmuchlater,intheearly1960s,thattheeldactually attractedenoughattentiontoattain\criticalmass",aslightlyAmericanbiasedpopular sciencerecountofthehistoryofChaosTheorycanbefoundin[ 35 ]. ThisdissertationisnotaboutChaosTheoryassuch,butmoreaboutsomeofthe \features"ofChaosTheory.Weutilizemuchofwhatisdened,predicted,andexpected fromChaosTheorytoproposeawholenewrealmforcomputation.Theproblemisthat ChaosTheoryisaverycloselypackedtheory,witheachpartrelatingtosomeotherpart andthewholetothedetails(aconceptwhichisaresultofthetheoryaswell).Wewill presentthe\features"ofChaosTheorythatarenecessaryforprojectingtothereaderour resultsanddirectthereaderto:Devaney(1982)[ 36 ]forasolidmathematicalexploration ofChaosandtheoriginofoneofthemostaccepteddenitionsofChaos;Peitgen,Jugens andSaupe(1992)[ 37 ]foramorehandsondemonstrationwithheavyemphasisonfractals, andOtt(1993)[ 38 ]foramidpointapproachtoChaosTheory,withtheadditionofan expositionofQuantumChaos. 1.2.1LogisticMap,TopologicalTransitivityandPeriodThree Inthisintroductorychapterweprovideasomewhat\non-traditional"viewofchaos. Theapproachwetakeisoneclosertoamathematician`sratherthanaphysicist`s,since 15

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computation,ourmainconcern,ismoreofamathematicalconceptthanphysical.We presentchaosthroughina\loose"settheorycontext,thatiswetreattopologicalspaces, intervalsandevenstatevariablesassetsofpoints,wetrytoavoidany\linearizing"tool, likedierentiation,andanyrepresentationthatinvolvesdiscretizationofoursetofpoints, likestatisticalmanipulations.Atthesametimeourexpositionis\loose",sincewedo notgointoextremeformalmathematicalprecision,thatisweomitprovidingproofsand extensivedenitionsoftermsused.Wedirectthereadertotwoseminalsources,Principia MathematicaofWhiteheadandRussell[ 30 ]andKurtGodel`severimportantpaper, \Onformallyundecidablepropositionsofprincipiamathematicaandrelatedsystems" (1931)[ 31 ],forbothmorebackgroundinformationofourapproach,especiallythemissing mathematicaldetails,andmoreimportantlyforjusticationforourapproachandhowit relatestotheconceptofcomputation. Ourmain\tool"forpresentingournecessarypartsofChaosTheoryisthediscrete timeLogisticmap: F n +1 ( x n )= x n +1 = x n (1 )]TJ/F2 1 Tf0.99999 0 Td[(x n ) ; (1{1) where x isthestateofthesystem, n 1 isthetimestepand0 < canbethoughtofasa growthrate.Actuallytheoriginsofthisquadraticequationisfrompopulationdynamics, specicallyitismodelingthebehaviourofapopulationwithlimitedresources.Inthat context x representsthecurrentfractionofthepopulationwithrespecttothemaximum possiblesustainablepopulation( x =1).Initscontinuous-timeform,themodelcanbe solvedanalyticallyandgivesrisetothemostcommonsigmoidfunction,a\wellbehaved" functionofcontinuousgrowthandeventualsaturation 2 .Whenweconsiderdiscretetime 1 Weshalldropthesuperscriptforthecasesof n n +1,unlesswewishtoemphasise theuseofasingleapplicationof F 2 AtheorembyPoincareandBendixson[ 39 ]guaranteesthatcontinuoustimetwo dimensionalplanarsystemscannotexhibitchaos. 16

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though,forexample n denotingthecurrentgenerationofbeingsinourpopulation,we haveEquation 1{1 andthebehaviourof x isnolongersimplecontinuousgrowthleading toeventualsaturation;infactinthisdiscretecaseifthepopulationdoesmanagetoreach saturationitcollapsestozero. ForourpurposesweneedacloserlookatEquation 1{1 .Witharstglanceitis clearthefunctionisgovernedbythethreevariables x n ,eachinprincipleunbounded. Thereisnoneedthoughtogotosuchlengths,aswecanconneourselvestothedomains 0 x 1,0 < 4,0
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richnessthatthissimplemapprovides,atleastthreebehavioursareevident:collapseto zero,attractiontoanon-zeroxedpointandattractiontoaperiodicorbit. Figure1-1.BifurcationdiagramfortheLogisticmap.Eventualbehaviourofaninitial state x 0 underrepeatedapplicationsof F Thisstaticpictureofwhathappens,eventhoughextremelyrich,isageneralization ofthedetailsofthebehavioursof x .Forexampleitshouldbeclear,butitisnotclearly shown,thatthispictureis\almost"completelyinvarianttotheinitial x 0 ,thatiswhat happenstoastate x 2 [0 ; 1]happensto\almost"anyotherinitialstate,themaphas alocaleectthatisglobal,andvisaversa.Itisnotclearthoughwhathappensatthe onsetofchaos,when\almost"allpointsbehavethesamewayandatthesametimeall the\rest"ofthepointsbehaveinadierentway! Toseethedetailsinsidethebifurcationdiagramwetakeamore\dynamic"approach withrespecttothethreeavailablevariables.Startingwith n wewillconsiderthequestion \Wheredostatescomefrom?",wearenotgoingtoaddresstheissueof\timereversal", 18

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butrathertakeastep\backintime"andconsider: x 0 = x )]TJ/F5 1 Tf0.8264 0 Td[(1 (1 )]TJ/F2 1 Tf0.99999 0 Td[(x )]TJ/F5 1 Tf0.8264 0 Td[(1 )= F ( x )]TJ/F5 1 Tf0.8264 0 Td[(1 ) ; (1{2) where x 0 2 [0 ; 1]and0 < 4.Figure 1-2 showstwoplotsofthefunction F 1 4 each showinghowthefunctionevolvesineither\directionoftime". Figure1-2.ForwardandBackwardevolutionof F 4 .(a)Forwardevolution-Emptycircle, # ,marksanarbitraryinitialpoint x 0 ;wecantrackitsforwardevolutionby moving\up"(or\down")tomeetthefunction F 4 andthen\right"(or\left") backtothediagonal,eachfullcycleofthesetwostepsisequivalenttoasingle applicationof F 4 ;thusthefullcircle, ,marksthepoint F 5 4 ( x 0 )= x 5 .(b) Backwardevolution-Thestepsoftheforwardevolutioncanalsobereversed; i.e.fromanarbitraryinitialpoint x 0 ,emptycircle # ,wecanmove\left"and \right"(or\left"twice)tothetwopointson F 4 andthen\up"and\down" (or\down"twice)tomeetthediagonalonthepointsof F )]TJ/F5 1 Tf0.8264 0 Td[(1 4 ( x 0 )= f x + )]TJ/F5 1 Tf0.8264 0 Td[(1 ;x )]TJ0 -0.98861 Td[()]TJ/F5 1 Tf0.8264 0 Td[(1 g thetwoemptysquares, o ;andofcoursefurtherbacktothefourpointsof F )]TJ/F5 1 Tf0.8264 0 Td[(2 4 ,thefourfullsquares, n ,andsoon. BysolvingEquation 1{2 weobservethefollowing:bysetting x 0 = x )]TJ/F5 1 Tf0.8264 0 Td[(1 wendthe xedpointsat x 1 =0and x 1 =1 )]TJET0.504 w265.39 148.89 m270.5 148.89 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm33.354 18.892 Td[(1 ;nowsetting x 0 =0and x 0 =1 )]TJET448.85 148.89 m453.96 148.89 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm56.369 18.892 Td[(1 wendforeach xedpointitstwopre-images,specically x )]TJ/F5 1 Tf0.8264 0 Td[(1 2f 0 ; 1 g! 0and x )]TJ/F5 1 Tf0.8264 0 Td[(1 2f 1 ; 1 )]TJET476.5 124.99 m481.61 124.99 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm59.84 15.892 Td[(1 g! 1 )]TJET529.27 124.99 m534.38 124.99 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm66.46 15.892 Td[(1 ; 19

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andnallybysolvingthequadraticforany x 0 wereach x )]TJ/F5 1 Tf0.8264 0 Td[(1 = 1 2 p 2 )]TJ/F5 1 Tf0.8264 0 Td[(4 x 0 2 ,whichleads to 4 4 ;F )]TJ/F5 1 Tf0.8264 0 Td[(1 ( x ) 2 C ;thepre-imagesofany x valuegreater thanafourthofthe\growthfactor"( ) ; oftheappliedmap,areinthecomplexplane. Thisisaninterestingresultandthestartofanotherlongstory,itshowshowthedynamics ofthelogisticmapextendinthecomplexplane.Thisisbeyondourcurrentscope,butin passingwenotethatthelogisticmapisareductionofthequadraticmaps Q c ( z )= z 2 + c where z;c 2 C ,thesourceofthefamousMandelbrotandJuliafractals[ 40 ],thisisan initialhinttotheconceptofUniversality,whichwewilladdressinSection 1.2.2 Wearenowreadytobegintoinvestigatehow changesthebehaviourof x 2 [0 ; 1]. Wecannotexplainwhathappensateveryvalueof soweconsiderthechangebefore andafterapointoftransition.Clearlyat =1wehaveamajorchange,thexedpoint x 1 =1 )]TJET128.23 493.92 m133.34 493.92 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm16.148 62.174 Td[(1 entersthedomain[0 ; 1]andbecomesanattractivexedpoint,actuallythe \bigevent"atthis -valueisthatthetwoxedpointscrosspaths.Atthesametime0 becomesarepellingxedpoint;theseconclusionscanbedrawnbytakingtheabsolute valueofthederivativeofthemapatthexedpoints, dF ( x 1 ) dx ,astandardtechniquefor characterizationofdynamicalsystems.Thisisthoughalocallinearizationofthemap,we prefertoshowthis,andsubsequenttransitions,intermsofglobaleects,thuspreparethe groundforwhathappensathighervaluesof Weneedtodeviateforawhiletosingleoutsomespecialpointsonthe I =[0 ; 1] interval.Wealreadyseenthefourpointsthatmaptothexedpoints, f 0 ; 1 ; 1 ; 1 )]TJET501.84 302.61 m506.95 302.61 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm63.02 38.177 Td[(1 g andweaddtothislist f 1 4 ; 1 2 ; 3 4 g .Ingeneral 1 2 isthemostimportantpointof F ,not becauseofitsvalue,butsinceitisthecriticalpointofthemapanditsbackwardand forwardevolutioncanactuallycharacterizethemapextensively;thetheorybehindthe evolutionofthecriticalpointiscalledkneadingtheory[ 36 ].Theimportanceofthe points 1 4 and 3 4 istopological,following F ( 1 4 ) F ( 1 2 )= 3 4 andofcourse F ( 1 4 )= F ( 3 4 ), inadditionto F ( 1 2 )=sup F ( I ),visuallythismeanswhatwillhappentotheinterval [ 1 4 ; 3 4 ] F n +2 ([ 1 4 ; 3 4 ])iswhathappenedpreviouslytotheinterval[ 3 4 ; 1] F n +1 ([ 3 4 ; 1])and 20

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whatisgoingtohappento[ 3 4 ; 1] F n +1 ([ 3 4 ; 1])iswhathappenedto[0 ; 1 4 ] F n +1 ([0 ; 1 4 ]), inanormalizedsense.Considertheinterval[0 ; 1]separatedinthreeparts I 1 =[0 ; 1 4 ), I 2 =[ 1 4 ; 3 4 ], I 3 =( 3 4 ; 1].Nomatterthe -valueweknowthefollowing: F (0)=0, F ( 1 4 )= F ( 3 4 )= 3 16 F ( 1 2 )= 1 4 and F (1)=0so F ( f 0 ; 1 4 ; 1 2 ; 3 4 ; 1 g ) !f 0 ; 3 4 ( 1 4 ) ; ( 1 4 ) g divideoutbyafactorof( 1 4 )anditslikenothinghappened!Weneedtoemphasisethat thispictureisonlyforvisualizationpurposesanditappliesonlytoasingleapplicationof themap,a\snapshot"ifyoulikethatwecankeeptrackofwhileiterating n !1 ,below wewilldiscusstheproperwaytoviewevolutionofintervals,therenormalizationoperator. Returningtothequestion\Whathappensas < 1 1 < ?".Therearemultiple wayswecananalysethesituation,welookatitwithrespecttochangestothebehaviour oftheaforementionedintervals.Globallybeforethepointoftransition( < 1)wesimply have F ( I ) < I soitsclearthatinthelimit n !1 thewholeintervalwillcollapseto zero.To\see"thechangeatthetransitionconsider F ( I 1 ) I n +1 1 [ I n +1 2 ,thatis I 1 will becometheunionofwhatcanbeconsideredtobethe I 1 and I 2 ofthenextapplication ( n +1)ofthemap,atwhich F n +1 ( I n +1 2 ) I n +2 3 and F n +2 ( I n +2 3 ) I n +3 1 [ I n +3 2 completingthe\circle".Actuallyitisadoublespiralwithamovingpivot,thesequence ofpost-imagesof 1 4 and 3 4 denotedas x .Sowehaveat < 1, F ( 1 2 ) < 1 4 ,whichimplies that F n ( I 2 ) \ I 2 = ,thisisenoughinsomesensetocharacterizetheevolutionsincewe haveseenthatinsomesenseboth I 3 and I 1 endup,atleastpartly,inanintervalthatcan becalled I m 2 ,anotherwaytodescribethisevolutionis F n ( x )=0as n !1 .Withthis descriptionoftheevolutionof F for < 1wecanseewhatchangesat =1,specically F ( I 2 ) \ I 2 6 = andatthesinglepointmagnication F 1 ( 1 2 )= 1 4 .Theconsequenceofthis changecanbeexpressedinmultipleways:sincethepost-imagesof I n 1 and I n 3 arein I n +1 2 itmeansthattherelationship F ( I 2 ) \ I 2 6 = canbeextendedto F n ( I 2 ) \ I n )]TJ/F5 1 Tf0.8264 0 Td[(1 2 6 = evenas n !1 so I 1 2 containsasinglepoint,whichistheeventualpost-imageofallthese pointsthatget\trapped"inthesequenceof I m 2 ;orsimplyas n !1 F n ( x ) 1 )]TJET516.74 163.36 m521.86 163.36 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm64.89 20.704 Td[(1 (=0,for =1);andeventhoughthe\pivot"pointdoesconvergetozeroeventually,there 21

PAGE 22

willalwaysbeatleastonemore\point"betweenitsvalueand0,these\points"wewill seearecentaltowhatischaos.Sonowanincreasein topplingitover1separates0and 1 )]TJET0.504 w94.32 713.23 m99.432 713.23 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm11.888 89.696 Td[(1 byamore\concrete"amount,givingsubstancetothexedpointat1 )]TJET481.68 713.23 m486.79 713.23 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm60.494 89.696 Td[(1 ,andto thepointsinsidetheinterval(0 ; 1 )]TJET261.29 689.32 m266.4 689.32 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm32.837 86.697 Td[(1 ),foraviewoftheevolutionofpointsineitherof thesetworegions( < 1or1 < )seeFigure 1-3 .Wewillcomeacrosstransitionslikethe onewejustdescribedaninnitenumberoftimesgoingfrom =1to =4. Figure1-3.Indicativebehaviourof I undermultipleapplicationsof F for,(a) < 1and (b)1 < .(a)( = 1 2 )Thetwoxedpointsareoneat0,andtheotheroutside I .(b)( =1 1 2 )0remainsaxedpointandnow1 )]TJET404.64 302.68 m409.75 302.68 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm50.826 38.181 Td[(1 2 I ;withonepre-image beingitselfat1 )]TJET228.82 286.12 m233.93 286.12 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm28.764 36.109 Td[(1 = 1 3 andtheotherat( 4 < ) 1 = 2 3 .Emptycircles, # ,mark initialpointsandfullcircles, ,nalpoints(notthexedpoint). Nowweareinthe1 < regionand,eventhoughthebifurcationdiagramdoesn't showit,thenextpointoftransitionisat =2.Wehaveseenhowthebehaviourof thewholeinterval I changedat =1,andwithitweshouldchangeourviewofits subintervals.Wecouldcontinuethediscussionwith I 1 I 2 and I 3 ,andwewillreturnto themwhennecessary.Ouraimthoughistomakethebehaviourofthewholeinterval I as 22

PAGE 23

visualaspossible,andsogiventhatnow1 )]TJET0.504 w307.01 761.11 m312.12 761.11 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm38.576 95.696 Td[(1 existsin I (aswellasbothitspre-images f 1 )]TJET101.02 737.2 m106.13 737.2 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm12.733 92.696 Td[(1 ; 1 g )wehaveabetterchoiceofsubintervalstouse,specically ~ J 1 =[0 ; 1 )]TJET500.83 737.2 m505.94 737.2 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm62.898 92.696 Td[(1 ), ~ J 2 =[1 )]TJET128.95 713.23 m134.06 713.23 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm16.238 89.696 Td[(1 ; 1 ], ~ J 3 =( 1 ; 1].Aswehavedonefortheprevioustransition,rstwewill seehowtheinterval I behaveswith1 << 2.Aquickside-note,takingthederivative atthexedpointswendthe\nature"of0toberepellingand1 )]TJET426.38 665.42 m431.5 665.42 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm53.557 83.697 Td[(1 attracting.The detailsthoughmakeallthedierence,aswehavedenedtheintervals ~ J i weseethat F 2 ( ~ J 0 3 )= ~ J 1 1 = F 1 ( ~ J 0 1 ).Theinterval ~ J 3 getsmappedonto ~ J 1 and ~ J 1 ontoitself,soitseems that ~ J 1 isnotchangingandevenmoreitseemsthat ~ J 1 and ~ J 3 are\disconnected" 4 from ~ J 2 .Soitseemswehavethreetypesofpoints,thesetoffourpoints f 0 ; 1 )]TJET457.7 569.8 m462.82 569.8 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm57.484 71.698 Td[(1 ; 1 ; 1 g ,andthe twointervals(1 )]TJET167.83 545.9 m172.94 545.9 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm21.111 68.699 Td[(1 ; 1 )and(0 ; 1 )]TJET255.96 545.9 m261.07 545.9 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm32.173 68.699 Td[(1 ) [ ( 1 ; 1),theeasypartisthexedpointsandtheir pre-images F ( f 0 ; 1 g )=0and F ( f 1 )]TJET277.42 522 m282.53 522 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm34.865 65.699 Td[(1 ; 1 g )=1 )]TJET346.46 522 m351.58 522 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm43.523 65.699 Td[(1 andinaddition,since 4 < f 1 ; 1 g weknowallotherpre-imagesofthexedpointsareinthecomplexplane.For(1 )]TJET504.94 498.09 m510.05 498.09 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm63.41 62.699 Td[(1 ; 1 ) since F :(1 )]TJET152.14 474.19 m157.25 474.19 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm19.145 59.699 Td[(1 ; 1 ) (1 )]TJET222.7 474.19 m227.81 474.19 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm28.002 59.699 Td[(1 ; 1 4 )and1 )]TJET310.68 474.19 m315.79 474.19 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm39.041 59.699 Td[(1 < 1 4 < 1 weseethatallpointsinthe limit n !1 are\squeezed"closerandcloserto1 )]TJET346.54 450.28 m351.65 450.28 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm43.533 56.7 Td[(1 .Finallyfor(0 ; 1 )]TJET456.7 450.28 m461.81 450.28 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm57.36 56.7 Td[(1 ) [ ( 1 ; 1),the simplepictureisthat 8 x 2 (0 ; 1 )]TJET251.35 426.38 m256.46 426.38 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm31.595 53.7 Td[(1 )and 8 y 2 ( 1 ; 1), F n ( x ) >x soas n !1 ;x 1 )]TJET529.27 426.38 m534.38 426.38 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm66.457 53.7 Td[(1 andaswehaveseen 8 x : F ( y )= x ,so F n +1 ( y ) 1 )]TJET365.04 402.48 m370.15 402.48 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm45.861 50.7 Td[(1 ,thereforeeverypointmoves awayfromzero(orone)andtowards1 )]TJET287.06 378.5 m292.18 378.5 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm36.074 47.701 Td[(1 .Wehavethoughseenthat F ( ~ J 1 )= ~ J 1 ,note that 8 x 2 ~ J 1 ;x< 4 soeverypointin ~ J 1 hastworealpre-images,oneofthemisofcourse in ~ J 3 andithasnopre-imagesofitsown,buttheotheroneisin ~ J 1 and x )]TJ/F5 1 Tf0.8264 0 Td[(1 x 0 )andeventually,at n !1 ,to1 )]TJET471.53 282.88 m476.64 282.88 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm59.217 35.702 Td[(1 ,buton everyapplicationofthemapitis\replaced"bytwootherpoints;inotherwords,points fromtheneighbourhoodof0,andofcourse1,eventuallyescapethisneighbourhoodand evolveintotheneighbourhoodof1 )]TJET266.69 211.17 m271.8 211.17 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm33.518 26.703 Td[(1 ,0isarepellingxedpointand1 )]TJET463.25 211.17 m468.36 211.17 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm58.174 26.703 Td[(1 attracting. 4 \Disconnected"hereisusedasineverydaylanguage,infactsince F n ( ~ J 3 ) 1 )]TJET0.504 w519.34 149.83 m524.45 149.83 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm65.214 19.004 Td[(1 as n !1 theintervalsareofcoursestillconnected. 23

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Assoonaswecrossfrom < 2into2 < weshallseemoreclearlytheimportancein consideringthisbehaviour. At =2wereachthesecondtransitionpoint,1 )]TJET0.504 w364.82 713.23 m369.94 713.23 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm45.83 89.696 Td[(1 = 1 = 1 2 ,insomesenseas at =1wehad1 )]TJET186.48 689.32 m191.59 689.32 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm23.457 86.697 Td[(1 =0,twoofthepointsthatleadtoaxedpointcrosspaths; thedierenceinthiscaseistwopointsthatleadtothesamexedpoint(1 )]TJET474.77 665.42 m479.88 665.42 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm59.619 83.697 Td[(1 and 1 ). Beforewegoontotheexcitingchangesthathappenin2 < ,wenoteinpassingthatat =2thederivativeatthexedpointis0andso 1 2 iscalled\super-attractive"point; 1 2 hasnopre-imagesrealorimaginary,inasenseallpoints(evenfromthecomplexplane) convergeto 1 2 ;moreimportantlythough,itistheonlypointin ~ J 2 asaresultwenowhave, inadditionto F 2 ( ~ J 3 )= F 2 ( ~ J 1 )= ~ J 1 F 2 ( ~ J 2 )= ~ J 2 ,thatisallourintervalsare\eventually invariant" 5 ,seeFigure 1-4 (a)foraviewofthebehaviourat =2. Wearenowinthe2 < region;therearetwoimportantgeneralchangesin behaviour,rstofthexedpointacquiresnowaninnitenumberofpre-images,since 1 < 4 ,andasaconsequence,topological\mixing"isnowpossible,thatisanyclosed intervalcontainspointsthathavepre-images,orevenpost-images,outsidetheinterval, thesetofpointsthatleadtothexedpointhasmorethanonepoint;morevisuallynotice that ~ J 1 \ ~ J 2 \ ~ J 3 6 = ,seeFigure 1-4 (b).Thisleadsustotherstconceptthatisa requirementforchaos,topologicaltransitivity. Denition1.1. TopologicalTransitivity. Givenamap f : J J f issaidtobetopologicallytransitiveon J ,ifforanytwo opensubintervals U V J thereexistsa x 2 U anda y 2 V suchthat f n ( x )= y ,for some 0
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thatwehaveintervals\mixing"istherststep,itisthedierencebetween\eventually invariant"andinvariant.Beforeweproceedletusredeneourintervalssothattheyare not\mixed"beforeweevenapplythemapandtokeepthepictureofhowtheintervals evolveasithasbeenuptonow.Sowere-dene J 1 =[0 ; 1 ), J 2 =[ 1 ; 1 )]TJET447.05 689.32 m452.16 689.32 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm56.148 86.697 Td[(1 ], J 3 =(1 )]TJET516.1 689.32 m521.21 689.32 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm64.809 86.697 Td[(1 ; 1], basicallyacknowledgingthefactthatthepre-imagesofthexedpoint(1 )]TJET465.55 665.42 m470.66 665.42 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm58.469 83.697 Td[(1 )have switchedsidesaround 1 2 ,seeFigure 1-4 (b)forthe\mixing"ofthe ~ J i intervalsandtheir redenitionas J i ,where i 2f 1 ; 2 ; 3 g .Thebehaviouroftheseintervalsisstraightforward, westillhave F ( J 3 )= F ( J 1 ),butnow F ( J 1 )= J 1 [ J 2 and F ( J 2 )= J 6 f 1 )]TJET494.14 593.71 m499.25 593.71 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm62.05 74.698 Td[(1 g[ J 3 Visuallywehave J 3 stretchedandrotatedaround1 )]TJET351.07 569.8 m356.18 569.8 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm44.103 71.698 Td[(1 onto J 1 [ J 2 J 1 issimplystretched onto J 1 [ J 2 and J 2 foldedinsideitself,with 1 2 asthepivotofthefolding,andagainrotated around1 )]TJET133.78 522 m138.89 522 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm16.839 65.699 Td[(1 tolandinside J 3 ;overallallpointsarespirallingaroundandinwards1 )]TJET522.5 522 m527.62 522 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm65.616 65.699 Td[(1 whichisthesituationwehadat < 1,soitshouldcomeasnosurprisewhathappensat =3. Figure1-4.Plotsof F 2 and F 2 : 5 .(a)Displayofthe\super-attractive"casewhen 1 2 isthe xedpoint.(b)Displayofthesixintervals f ~ J 1 ; ~ J 2 ; ~ J 3 g and f J 1 ;J 2 ;J 3 g ,the overlapof ~ J i showstheinitiationoftopological\mixing". 25

PAGE 26

Asthepointsarespirallinginwardsthexedpoint,and isgraduallyraisedwehave in J 3 whatwashappeningto I ,thepointscollapsingtowardsthexedpoint,untilthereis apointleftout,specicallywhen F ( 1 2 ) (1 )]TJET319.9 713.23 m325.01 713.23 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm40.194 89.696 Td[(1 )+ 1 4 h 1 )]TJ/F0 1 Tf0.99999 0 Td[((1 )]TJET414 713.23 m419.11 713.23 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm52 89.696 Td[(1 ) i 6 ;ormorevisually whenthestretchingof J 3 (and J 1 )\creates"morepointsthanthe\available"pointsin J 1 [ J 2 andso\a"pointisleftoutthatwillneverspiralon1 )]TJET403.99 665.42 m409.1 665.42 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm50.74 83.697 Td[(1 7 .Allthishappensat =3and,asat =1,thepointleftoutisapost-imageof 1 2 onlythistimewehave,for 0
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L ( y )= ( y )]TJ/F3 1 Tf0.8264 0 Td[(x 1 ) ^ x 1 )]TJ/F3 1 Tf0.8264 0 Td[(x 1 and L )]TJ/F5 1 Tf0.8264 0 Td[(1 ( y )= x 1 +(^ x 1 )]TJ/F2 1 Tf0.99999 0 Td[(x 1 ) y Wehave: RF ( x )= L )]TJ/F2 1 Tf0.45834 -0.81001 Td[(F 2 )]TJ/F2 1 Tf0.45833 -0.81001 Td[(L )]TJ/F5 1 Tf0.8264 0 Td[(1 ( x ) = L F 2 L )]TJ/F5 1 Tf0.8264 0 Td[(1 ( x ). Where RF istherenormalizedfunctionof F 2 thattranslates J 2 =[ 1 ; 1 )]TJET484.7 689.32 m489.82 689.32 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm60.866 86.697 Td[(1 ]onto I =[0 ; 1],bothbyreectingitupwardsandrotatingaroundsothat1 )]TJET445.46 665.42 m450.58 665.42 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm55.944 83.697 Td[(1 =0and 1 =1, seeFigure 1-6 fortwoexplicitexamplesofrenormalization,relatingtotheissuesof < 0 and4 < .Thisshouldmakeitclearhoweverythingthathappenedto F 1 willhappen to F 2 ;ofcourseappropriate L and L )]TJ/F5 1 Tf0.8264 0 Td[(1 canbefoundforall n ,butthisisbeyondour currentscope,seetheworkofKennethG.Wilson[ 41 42 ].Thissidetrack 8 wasintended toshowhowtheactualvalueof isvery\relative",aswellastheactualintervalof x ;or tobemoreprecisetheactualsetof x values. Returningtothe3 < regionandhowthecomplexityofbehavioursincreases exponentiallywithincreasing .Atthemomentwehavetwoattractingxedpoints,each withitstwopre-images,oneofthembeingtheotherxedpointandtheotherpre-image hasitsowntwopre-imagesandsoon.Theprocesswehaveexploreduptonowwill repeat,infactinnitetimes,as israisedfurther.Wewillhavea\super-attractive"case, when F 2 ( 1 2 )= 1 2 ,at 3 : 23 ::: ,andofcoursethetrappingofyetanothertwopointsfor eachoneofthe\current"twocyclexedpoints,andthusthecreationofthefourxed pointcycle,at =1+ p 6 3 : 44 ::: andofcoursethechangeofthetwocyclexedpoints intorepellingpoints.Thefourcyclewillbecomeeightcycleandsoonforallpowersof2. Toourvisualpicture,westillhavetheactionoftheprimordialxedpoint,1 )]TJET484.42 282.88 m489.53 282.88 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm60.835 35.702 Td[(1 ,even thoughnow,at3 < itisrepelling,itspivotactioninthespiraling-instillexists.In additionwehavemore\local"eectsfromthenewbornxedpointsactingaspivotsfora spiralling-inoftheirlocalsurroundingpoints.Ineectwehavevorticeswithinvortices. 8 Anotherreasonforthisdiversionistoprovideasmallglimpse,withoutgoinginto details,tothepictureof F 2 k 1 with k !1 ,whichwewillmeetsoon. 27

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Figure1-5.Exhibitionofthe\trapping"oftwopointsintwodierentcongurationsof F .(a)Plotofboth F 1 3 : 2 ,solidline,and F 2 3 : 2 ,dashedlineshowingtheeventual evolutionofdierentinitialpoints,markedby # ,tothe2-cyclexedpoints markedby .Thesolidtriangles, K ,markthesequencebetweenthetwoxed points.Notethat F 2 andthediagonalcanbeusedlike F 1 inFigure 1-2 to trackthe2 n -stepevolutionofpoints.(b)Plotof F 1 2+ p 5 aswellasthe evolutionsofsomedierentinitialpoints,markedby # ,undermultiple applicationsof F 2+ p 5 .Notethatthetwopointsmarkedby ,are\trapped" by F 1 2+ p 5 ,whiletheirfourpre-images,markedby n ,are\trapped"by F 2 2+ p 5 andsoonto F 1 2+ p 5 ,andevenfurtherto F 2+ p 5 2 1 Asmoreandmorexedpointsarecreatedthegapsbetweenthemforallthisswirling totakeplacearegettingsmallerandsmaller,leadingtosmallerandsmallerincrements of tocreatethesubsequentdoubling,wewillreturntothisevolutioninSection 1.2.2 whenwetalkaboutUniversality.Fornowweareinterestedinwhathappensatthe accumulationpoint,alsoknowasFeigenbaumpoint,namedafteritsdiscovererMitchell Feigenbaum, 1 3 : 569945671 ::: .Letusattempttoaccountallthepointsin I andtheir behaviourstartingwiththeeasiestpart f 0 ; 1 g thetwopointsthatcollapseto0.Wehave thexedpointof F 1 ,thatis1 )]TJET0.504 w244.15 141.26 m249.26 141.26 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm30.69 17.936 Td[(1 ,withitwestillhaveitspre-image 1 and\goingfurther 28

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Figure1-6.Renormalizationoftwocasesof F 2 .(a)Caseof F 2 3 : 8 ,forwhich L 3 : 8 F 2 3 : 8 L )]TJ/F5 1 Tf0.8264 0 Td[(1 3 : 8 ( x )translatesinto(b)Aplotof RF ( x ),showinghow representativepoints,markedbysymbols,aretranslatedontopointsofsimilar topology.Itshouldalsomakeclearhowcasesof4 < are\contained"insome formin0 < 4.(c)Caseof F 2 3 ,forwhich L 3 F 2 3 L )]TJ/F5 1 Tf0.8264 0 Td[(1 3 ( x )translatesinto (d)Aplotof RF ( x ),showinghowrepresentativepoints,markedbysymbols, aretranslatedontopointsofsimilartopology.Itshouldalsomakeclearhow casesof < 0are\contained"insomeformin0 < 4. backtime"allthepointsthatleadtoit,aswementionedinnitenumberofthem.From F 2 wehavethetwoxedpoints,ofcoursedierentfrom1 )]TJET0.504 w388.58 316.94 m393.7 316.94 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm48.81 39.972 Td[(1 and 1 ,eachofthesetwo xedpointshastheotheroneasoneofitspre-images,butinadditioneachpointhasone morepre-image,whichofcoursehasitsownpre-images,onceagainininnitenumber. Continuingthisforeach F n with n =2 k as k !1 wecanaccountforallxedpointsof periodicityapowerof2,asalreadyexplainedabovethroughtheperiod-doublingroute, andalltheirpre-images.Withallthesepointsaccountedfor,itmightseemwehaverun outofpointsfor F 2 k 1 at k !1 ,tohaveanyxedpoints,infactthoughwestillhavejust asmanypointsasweaccountedfor,wehaveaCantorsetonceagainandtechnicallyour stinstanceofchaos.ThedetailsofthebehaviourattheFeigenbaumpoint( 1 )of F 2 1 1 29

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andforthatmatterof F 1 1 ,arecomplicatedtodescribe,bothvisuallyandmathematically, withouttheintroductionofsymbolicdynamics,furtherexpositionofCantorsetsand renormalizationgroupoperator,conceptswhicharebeyondourcurrentscope.Instead wewillproceedinincreasing furtherandshowhow F 1 becomeschaoticonthewhole interval I =[0 ; 1]at =4,atwhichpointthebehavioursarethesameasfor F 2 1 1 at 1 buteasiertodescribeandvisualize. WeutilizetheperioddoublingroutetotheFeigenbaumpointtointroducethesecond requirementforchaos,xedpointsofperiodicitythree.Theoriginaldenitionachaotic systembyDevaney[ 36 ]requiredofasystem:(i)TopologicalTransitivity,(ii)thesetof periodicpointstobedenseinthespaceofthesystem,and(iii)sensitivedependanceon initialconditions.Bankset.al.[ 43 ]though,showedhowtransitivityanddensityofthe setofperiodicpointsimplysensitivedependanceoninitialconditions,whileLiandYorke [ 44 ]provedthatexistenceofperiodthreexedpointimpliesthesetofperiodicpointsis dense;infacttheyalsoshowhowthesetof\non-periodic"pointsisalsodense.BeforeLi andYorkethough,itwasSharkovsky[ 45 46 ]thatintroducedhisfamoustheorem 9 that actuallyclaimsmorethanjustperiodthreeimpliesallotherperiodicities,itactually providesanorderingofthenatural 10 numbersthatwillguideustondxedpointsof periodicitiesotherthanapowerof2. Theorem1.1. Sharkovsky'sTheorem. Givenacontinuousmap f : R R ,theorderingofthenaturalnumbers: 1 C 2 C 2 2 C 2 3 C ::: C 2 n C ::: C 7 2 n C 5 2 n C 3 2 n C ::: ::: C 7 2 C 5 2 C 3 2 ::: C 9 C 7 C 5 C 3 9 Eventhoughthetheoremappliesonlyformapsontherealline,itsstrongconclusion makesitimportanttonoteinanyexpositionofchaos. 10 ActuallytheorderingisformorethanjusttheNatural( N )numbers,itcoversas manyperiodicitiesasalltheReals( R )[ 47 ]. 30

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andifthemaphasaxedpointofperiodicity q ,where p C q ,thenthemapalsohas axedpointofperiodicity p .Ifthemapdoesnothaveaxedpointofperiodicity p ,where p C q ,thenthemapdoesnothaveaxedpointofperiodicity q .Further,forany p a continuousmapdoesexistwhichhasxedpointsofperiodicity p ,butnoxedpointsofany periodicity q ThemostobviousimplicationofSharkovsky'stheorem,whichisanaloguetothe Li-Yorketheorem,isthatifamaphasaxedpointofperiodthreethenithasxed pointsofallnumberperiodicities.Ofcoursethespecicationoftheorderingallowsusto saymore,ifamaphasanitenumberofperiodicpointsthentheyareallapowerof2, andtheconverseifamaphasaperiodicpointthatisnotapowerof2thenithasinnite numberofperiodicpoints.Thesecondpartofthetheoremallowsfortheexistenceof mapswithaninnitenumberofpointsofdistinctperiodicityandyetwithoutallnatural numberperiodicities.Actuallytheexistenceofsuchmapscanbeutilizedintheproofof thetheoremtoestablishthisnumberordering,andperhapsmoreimportantlytoguarantee thesuccessofthethresholdcontrolmechanismforchaoticsystems,seeSection 1.2.3 FurthermorethereisacorollarytoSharkovsky'stheorem,whichstatesthatifthe map f isdependantonavaryingparametertheorderingofthe\birth"ofnewperiod xedpointsisgivenbytheSharkovskyordering,whichbringsusbackto F andthe furtherchangesasweincrease .Wehavealreadyseenthebirthofallperiodsofpower of2uptotheFeigenbaumpoint,ournextstepistoshowthebirthofallotherxed pointsofevenperiodicitybyincreasing overtheFeigenbaumpointto 2 3 : 67 ::: 11 and showinghowthisregionof allowsonlyevenperiodicities. Readjustingtheintervalsunderobservationwedene J 4 =[ F 2 ( 1 2 ) ;F ( 1 2 )],ormore clearlyusing F h and h ( 1 2 )= 4 J 4 =[ h ( 4 ) ; 4 ];inadditionwedene J 5 =[ h ( 4 ) ; 1 )]TJET526.1 211.17 m531.22 211.17 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm66.062 26.703 Td[(1 ] 11 Thealgebraicexpressionfor 2 = 2 3 [ 4 u + u +1],where u =(19+3 p 33) 1 3 ;obtainedeither by F 2 ( 1 2 )=1 )]TJET155.3 135.36 m160.42 135.36 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm19.543 17.192 Td[(1 orsup f F )]TJ/F5 1 Tf0.8264 0 Td[(1 ( 1 ) g 4 31

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and J 6 =[1 )]TJET0.504 w149.76 761.11 m154.87 761.11 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm18.847 95.696 Td[(1 ; 4 ],thereforewehave F ( J 5 ) J 6 and F ( J 6 ) J 5 ,for 1 << 2 Beforefocusingontheseintervalsletusconsidertherestofthepointsin I ,ofcourse F ( f 0 ; 1 g )=0nothingnewthere; F :( 4 ; 1) )]TJ/F0 1 Tf0.45833 -0.81001 Td[(0 ;h ( 4 ) and F n : )]TJ/F0 1 Tf0.45834 -0.81001 Td[(0 ;h ( 4 ) J 4 as n !1 .Soitisclearthatallpointsin I eventuallyaremappedinside J 4 ,once therewehave :::J 5 J 6 J 5 ::: soitisimpossibletohaveapointthatismapped backonitselfafteranoddnumberofapplicationsofthemap,furthermorepointsofall evenperiodicitiesarepossibleandexist.At 2 wecanseewhathappensto F 2 andit ispreciselywhathashappenedtoeachotherevenperiodicitywhileincreasing from theFeigenbaumpoint,withasmallbutimportantdierenceinthebehaviourof 1 2 Specically F 2 2 ( 1 2 )= 1 thatis 1 2 ismappedtothepre-imageoftheprimordialxed point,so 1 2 isapre-imageofaxedpoint,butnotitselfxedasinthe\super-attractive" case.Visuallythismeansthatthereisatleastoneintervalthatiscompletelywrapped aroundontoitself,incontrasttopreviouslywhen 1 2 was\eventually"mappedontoaxed point.Wecanalsonowseethedierencebetween\eventuallyinvariant"andinvariant, F 2 2 ( J 5 )= J 5 and F 2 2 ( J 6 )= J 6 ,notetheequality,andseeFigure 1-7 (a)foravisualof theintervals.Perhapsmostimportantlybothpre-imagesof 1 arenowlessthan 4 ,which impliesthatbothhavetworealpre-images,andatleasttwoofthosepre-imageseachhave twopre-imagesoftheirown,andsoon;thexedpoint1 )]TJET379.44 354.6 m384.55 354.6 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm47.665 44.701 Td[(1 hasnowasetofpre-images whichformaCantorset.Whenweputalltheseimplicationstogether,alongwiththefact thattheSharkovskyorderingfor F 2 isthesameastheorderingfor F 1 withoutthelastleg oftheoddintegers(soallnumberperiodicitiesexistfor F 2 2 ),itshouldbeclearthat F 2 2 is chaotic.Itisimportanttonotethat F 2 2 ischaotic on J 4 notthewhole I 12 ThereforenowweareleftwiththelastlegoftheSharkovskyordering,theodd numberperiodicities.Obviouslythemostimportantbeingperiodthreeandtheeasiest onetovisualizesinceitisrelatively\short".Butbeforewefocusonperiodthreeletssee 12 Actually F 2 isindependentlychaoticon J 5 and J 6 32

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howoddperiodicitiesbecomepossibleintherstplace.Soat 2 < wehaveseenthat F 2 2 ( 1 2 ) < 1 ,so F 2 ( J 2 ) \ J 1 6 = ? ortorelatetwosetsfromdierentviews J 4 \ J 1 6 = ? ,lets callthisintersection J 7 J 5 ,mostimportantly J 7 < 1 ) F ( J 7 ) < 1 )]TJET446.04 713.23 m451.15 713.23 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm56.016 89.696 Td[(1 ) F ( J 7 ) J 5 sowehave J 7 J 5 ::: J 6 J 5 ::: | {z} infinite 9 eventimes J 6 J 7 ,hencethe\highest"odd periodicityorbitis\born",seeFigure 1-7 (b). Figure1-7.Demonstrationofbirthofoddperiodicityxedpoints.(a)Plotof F 2 ,solid line,and F 2 2 ,dashedline.Wealsomarktheintervals J 4 J 5 and J 6 ,itshould beclearthat F 2 : J 5 J 6 and F 2 : J 6 J 5 ; F 2 2 ischaoticonboththese intervals.Notealsothat F 2 : I f 0 ; 1 g J 4 J 4 .(b)Plotof F ,solidline, and F 2 ,dashedline,at = 3 +0 : 02,avalueforwhichtheperiodthreexed pointsareattractive.Notehow J 5 \ J 1 = J 7 6 = ? .Within J 7 wehaveoneof theperiodthreetrappingregions,seeinset.(c)Inset,showingaplotof F 3 ( x ), for = 3 +0 : 02and x 2 [0 : 1358 ; 0 : 1722],topologicallythesameassome F ( x ),for x 2 I 9 Ornoneinthecaseofperiodthree. 33

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Wereturnnowtotheissueofthebirthofthexedpointsofperiodicitythreefor F 1 At 3 =1+2 p 2 14 wehavethebirthoftheperiodthreeorbit,thesimplestwaytoview itisthatthesolutionstotheequation F 3 )]TJ/F2 1 Tf1.0353 0 Td[(x =0,becomerealfromcomplex.Interms ofourvisualpicture,wehaveagrowthfactorwhichislargeenoughfor 1 2 tofalloutside the\trapping"region.Thesituationisverysimilartowhathappenedat =1whenwe hadtheintroductionofa\new"xedpointin I by F 1 ( I 2 ) \ I 2 6 = ,onlythistimethe setofthisintersectiondoesnotcontainapost-imageof 1 2 F 1 ( 1 2 ) = 2 F 1 ( I 2 ) \ I 2 ,sothisset containssomeotherpointormorecorrectlypoints.Sincenowitisnotthemaximalpoint thatgetscaptureditmeansthattheintersectioncannotbecollapsedtoasinglepoint, buttherewillalwaysbeatleasttwopoints,markingtheboundariesofthisintersection interval,andsinceatleastboththesepointsare\trapped"wehavetwonewxedpoints, foreachtrappingregion.Onceweconsiderthatwearedealingwith F 3 ,therearethree \trapping"regions,soatotalofsixnewxedpoints,alongwiththetwoprimordial, f 0 ; 1 )]TJET111.1 450.28 m116.21 450.28 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm13.997 56.7 Td[(1 g ,eightintotalagreeingwiththefactthat F 3 isaneighthdegreepolynomial 15 Thenatureofthesepointsisoneofthemattractingandtheotherrepelling,sinceweare talkingaboutanintervalbeing\trapped"thepointsthatactuallyexistbetweenthetwo xedpointsarealso\trapped"byeventuallycollapsingtotheattractingxedpoint.So theoverallsituationinatrappingregionis\identical"tothesituationwehadin I for 1 < ,seeinset(c)ofFigure 1-7 .Thereforeitshouldbeofnosurprisethateverything thatwehavetalkedaboutuptonowfor I happenswithineach\trapping"regionoverand overagainasweincrease evenfurther;insomesensethisisthereasonwhyperiodthree implieschaosandnotsimplythefactthatitallowsallotherperiodicities;eventhoughone argumentimpliestheother. 14 Derivationofthisvaluecanbefoundin[ 48 { 50 ]. 15 Infactsince F isasecondorderpolynomialany F k willbeapolynomialoforder2 k sosolutionstoanycaseof F k )]TJ/F2 1 Tf0.99999 0 Td[(x =0,willhaveanevennumberofrealroots. 34

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Finally 3 < 4,with =4themostimportantpart,givenonFigure 1-8 .In theregionbetween 3 and4everythingwehaveseenrepeatsinnitenumberoftimes. Insomesubsetof I orother,wehavethecreationofnewotherxedpointsthatturn fromstabletounstablegivingbirthtomoreandmorexedpointsinthesameway,all thewaytothecreationofthe\periodthreeorbit",forthesubsetinquestion,when,of course,westartoveragain!At =4wehavethenalbrickonthewall, F 4 ( 1 2 )=1! 16 Wehaveseentheconsequencesofthisbefore,at 2 1 2 ismappedto0,theotherxed point,andasbefore,moreimportantly1 4 thepre-imagesof0formaCantorset;and afterthislongjourneywenallyhave F : I I F ischaoticonthewholeinterval I Wehaveseenallthathasbeencreatedasweincreased ,andeverythingisstillthere,so wewillnotrepeatourselves.Instead,andinconclusiontothissection,wewillprovide aholisticpictureofthepointsin I .Wehaveasetofpointswhicharexed,ofsome periodorother,thissetisdensein I ,thereisadicultissuehere,wecanexcludefrom thissetthexedpointsof F 2 1 17 ,westillhavethexedpointsofallperiodicities,even innity,butweareonlyconsideringtheenumerablenumberperiodicities,andthissetis stilldensein I .Thesetofxedpointsof F 2 1 wecannowputontheirownanditisin itselfaCantorset,andofcoursedensein I ,butevenmore,uncountable.Wearenotdone though,therearestill\many"pointsleftandspecicallythepre-imagesofallxedpoints ofenumerableperiodicity,whichaswehaveseenforeachxedpointformaCantorseton theirown.Hencewecanconsiderthefollowingtwosets,thexedpointsof F 2 1 (andtheir pre-imagesinsomesense),andthesetofallotherxedpointsandtheirpre-images;both setsareaninnitecollectionofCantorsets,bothdensein I ,bothuncountable.Forits 16 Subsequentlyweshalldropthesubscripton F 4 andrefertothefunctionofthis parametersimplyas F 17 Notethat hereisnottheFeigenbaumpointvalueof 1 ,insteadweareconsidering allpointsofall F 2 1 -functions,eventheonescreatedafterperiodthreerestartsthe process. 35

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immenseimportance,ournalpointinthissectionistoemphasisethefollowing,wehave seenhowafunctiononitsownisnotchaotic,asasetonitsownisnotchaotic,itisthe combinationofamany-onefunctiononaspecicset,underspecicparametersthatcreate chaos. Figure1-8.Plotofthefunction F : I I .Alsomarkedaretheintervals I 1 I 3 and I 2 ,the lastbrokenupinto I 2 L and I 2 R ,whicharethetwointervalsthataremapped completelyontoeachother. 36

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Table1-1.Summaryofthetransitionsinbehaviourofthedierentintervalsdescribedin Section 1.2 .Eacharrowrepresentsasingleapplicationof F x isusedto indicatethatmorethanonepointistrappedintheinterval,while y isusedto indicatethatasinglepointistrapped(typicallyanendpointoftheinterv al). I = I 1 [ I 2 [ I 3 I = ~ J 1 [ ~ J 3 I = J 1 [ J 2 [ J 3 I =( J 1 J 5 ) [ J 5 [ J 6 [ ( J 3 J 6 ) 0 1 x I 1 I 2 I 3 1 2 yx I 1 I 2 "% I 3 xx ~ J 1 ~ J 2 ~ J 3 2 3 yx I 1 I 2 "% I 3 xx ~ J 1 ~ J 3 ~ J 2 = ~ J 1 \ ~ J 3 yy J 1 J 2 "%. J 3 yy J 5 J 6 3 1 yx I 1 I 2 "%. I 3 xx ~ J 1 ~ J 3 ~ J 2 = J 1 \ J 3 yy J 1 J 2 "%. J 3 yy J 5 J 6 1 2 yx I 1 I 2 "%. I 3 xx ~ J 1 ~ J 3 ~ J 2 = ~ J 1 \ ~ J 3 yy J 1 J 2 "%. J 3 yy J 5 J 6 2 3 yx I 1 I 2 "%. I 3 xx ~ J 1 ~ J 3 ~ J 2 = ~ J 1 \ ~ J 3 yy J 1 J 2 "%. J 3 yy ~ J 5 J 6 ". J 7 x J 5 = ~ J 5 [ J 7 3 4 yy I 1 I 2 "%. I 3 xx ~ J 1 ~ J 3 ~ J 2 = ~ J 1 \ ~ J 3 yy J 1 J 2 "%. J 3 yy ~ J 5 J 6 ". y J 7 x J 5 = ~ J 5 [ J 7 Thetransitionshownactuallyoccursat =1 1 3 37

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1.2.2TheTentMap,TopologicalConjugacyandUniversality ThissectionwillbrieyintroducetheTentmap,whichalongwiththeLogisticmap arethetwomainsystemsweuseforchaosbasedalgorithmdevelopment.TheTentmap isapiecewiselinearmapthatalsocanexhibitchaos.Insteadofpresentingindetailthe chaosintheTentmap,aswehavedoneintheprevioussectionfortheLogisticmap,we willintroducetheconceptofTopologicalConjugacy,whichrelatesonemaptotheother, guaranteeingthepropertiesandbehavioursinonemaptoexistintheother.Inclosingthe sectionwewillmovefurtherthanTopologicalConjugacy,intothepromisedpropertyof chaosknownasUniversality.Universalityplaysaveryimportantroleincomputationwith chaoticsystemssinceitreleasesusfromconnementtoanysinglephysicalrealization;as longasasystemischaoticthealgorithmswedevelopcanbeimplementedbythesystem, regardlessofwhetherthenatureofthesystemiselectrical,optical,chemical,orevena realizationinsomeotherphysicalrealm. TheTentmap .JustastheLogisticmap,theTentmapisdiscretetime,butinstead ofa\continuous"polynomial,itisapiecewiselinear\discontinuous"mappingoftheform: T n +1 ( x n )= x n +1 = 8 > > < > > : x n ; for x n < 1 2 ; (1 )]TJ/F2 1 Tf0.99999 0 Td[(x n ) ; for 1 2 x n ; (1{3) where x 2 [0 ; 1]( I ),0
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Figure 1-10 ,eventhoughitlooksmore\compressed"andthebifurcationsfortheperiod doublingarecloselypacked,basicallyeverythingthatweshowedfortheLogisticmap appliestotheTentmapaswell.Wecanactuallyshowthis\identity"mathematically throughtheTopologicalconjugacyofthetwomaps. Figure1-9.Plotofthefunction T : I I .Fromthegraphwecanseehow T : I 1 I 1 [ I 2 T : I 3 I 1 [ I 2 ,and T : I 2 L I 3 and T : I 2 R I 3 ;andof coursehow 1 2 isapre-imageof0. TopologicalConjugacy .Topologicalconjugacyhasitsrootsinsettheory, considertwodistinctsets, X and Y ,andforeachsetthereisa\relation",say P and Q respectively,betweenitsindividualelements.Foreachsetconsiderthe\larger"setof theunionoftheelementsthatareinthedomainandconversedomainofthe\relation", 39

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Figure1-10.BifurcationdiagramfortheTentmap.Eventualevolutionofanarbitrary initialstate x 0 .Inset:Magnicationoftheregion 2 [1 ; 1 : 1]and x 2 [0 : 495 ; 0 : 5],showingmoreclearlytheperiod-doubling. the\eld"oftheset,forexamplegiven x 2 X : P ( x )= x ,then x and x areinthe\eld" of X and P ,analogouslyfor Y and Q .Ifthereisa\relation" S ,whichisone-to-oneand hasasdomaintheeld f X ; P : X g andconversedomain f Y ; Q : Y g ,thenthe\relations" P and Q are\similar",thatmeans P and Q haveidenticalproperties,inotherwords theireectsontheirrespectivesetsareindistinguishable.Whenthesetsconsideredare topologicalspaces,thecaseoftheLogisticandTentmapon I ,andthe\relations"actual functions,thenthefunctionsaresaidtobeTopologicallyconjugatetoeachother. Tomakethisconceptmoreconcrete,andshowaspromisedthatthereisnodierence betweentheLogisticandTentmaps,considerthetwofunctions F : I I and T : I I ; thefactthat I actsasthedomainandconversedomainforbothfunctionsshouldnot worryusfortworeasons,rstofthesets(orspaces)consideredareactuallyarbitrary 40

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Table1-2.Topologicalconjugacy.GiventhattheTentmaptakesevery x 2 I ona x 2 I andtheLogisticmaptakesevery y 2 I ona y 2 I ,theyareTopologically conjugatewhenafunction, G ,existsthattakesevery x 2 I ona y 2 I and every x 2 I ona y 2 I ;sothat G T ( x )= F G ( x )= y .Specically G ( x )=sin 2 )]TJET0.504 w197.21 705.16 m202.39 705.16 lSBT/F3 1 Tf7.9706 0 0 7.9706 0 0 Tm24.743 88.677 Td[( 2 x I T )166(! I ( x )( x ) G ## G I F )166(! I ( y )( y ) andinrealityunchanged 19 itisthe\relations"(orfunctions)thatwewillactually manipulate,andsecond,theactualsetofpointsin I foreachfunctionisdierent, bothfunctionsareappliedonthesame\interval",whichcontainsthesamepoints, butthepointsthemselvesaredierentforeachfunction,except f 0 ; 1 2 ; 1 g .Sofollowing Table 1-2 wehavethefollowingrelationships: T : I I F : I I G : I I andofcourseforany x;y; x; y 2 I ;T ( x )= x;F ( y )= y;G ( x )= y and G ( x )= y whichimply G T ( x )= G ( x )= y or F G ( x )= F ( y )= y .FortheLogistic andTentmaptheconjugatingfunctionisgivenby G ( x )=sin 2 )]TJET409.9 386.64 m415.08 386.64 lSBT/F3 1 Tf7.9706 0 0 7.9706 0 0 Tm51.428 48.718 Td[( 2 x ,specically: y 1 = F G ( x 0 ) y 1 =4 y 0 (1 )]TJ/F2 1 Tf0.99999 0 Td[(y 0 ) y 1 =4 sin 2 )]TJET152.35 301.96 m157.54 301.96 lSBT/F3 1 Tf7.9706 0 0 7.9706 0 0 Tm19.121 38.093 Td[( 2 x 0 )]TJ/F0 1 Tf0.45834 -0.81001 Td[((1 )]TJ/F0 1 Tf0.99999 0 Td[(sin 2 )]TJET251.14 301.96 m256.32 301.96 lSBT/F3 1 Tf7.9706 0 0 7.9706 0 0 Tm31.51 38.093 Td[( 2 x 0 y 1 =4 sin 2 )]TJET152.35 273.09 m157.54 273.09 lSBT/F3 1 Tf7.9706 0 0 7.9706 0 0 Tm19.121 34.469 Td[( 2 x 0 cos 2 )]TJET221.9 273.09 m227.09 273.09 lSBT/F3 1 Tf7.9706 0 0 7.9706 0 0 Tm27.847 34.469 Td[( 2 x 0 y 1 =sin 2 ( x 0 ) y 1 = G T ( x 0 ) x 0 < 1 2 1 2 x 0 x 1 =2 x 0 x 1 =2 (1 )]TJ/F2 1 Tf0.99999 0 Td[(x 0 ) y 1 =sin 2 )]TJET366.34 273.09 m371.52 273.09 lSBT/F3 1 Tf7.9706 0 0 7.9706 0 0 Tm45.963 34.469 Td[( 2 x 1 y 1 =sin 2 ( )]TJ/F2 1 Tf0.99999 0 Td[( x 0 ) y 1 =sin 2 ( x 0 ) y 1 =sin 2 ( x 0 ) 19 Youcanactuallyconsidereachdomainandconversedomainasxedpointsinaset andthe\relations"asjusta\virtual"linkbetweenthesepointsineachset,notactually aectingthepoints. 41

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Ofcoursebyinduction,ifnotbysimplyconsideringfunctioncomposition,thisresult appliesforany,andatany, n F n G = G T n ,seeFigure 1-11 foranillustration. Whenweconsiderthe\spatial"correlationbetweenfunctionsgivenbyTopological conjugacy,inadditiontothe\temporal"correlationbetweenafunctionanditsfuture iterates(e.g. F and F n )givenbyrenormalization,wehaveauniversalcorrelationbetween any\functions"thatexhibitsimilarbehaviours. Figure1-11.Topologicalconjugacybetweenevolvedstates,upto n =5.Thedashedline followstheevolutionof G ( x 0 )undertheLogisticmap( F ),whilethe dash-dottedlinefollowstheevolutionof T n ( x 0 ).Theactionof G ( x )bothas therststep( x 0 G y 0 )andlaststep( x 5 G y 5 )isshownalongthetriangular arrows.( f ( x )is f F (parabola), T (piecewiselinear), G (sigmoid), I (identity) g ). 42

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Universality .ThetheoryofUniversalityis,asthenameimplies,extremelywideand intricate.ItcorrelatesmanybranchesofPhysics,fromthetheoryofcriticalphenomena toHamiltonianmechanics,andmanybranchesofMathematics,fromstatisticstovector spaces.Theaimofthissectionistopresentsomeimportantqualitativeresults,both asahintofproofofUniversality,anditsmainconsequence,theindependencebetween \behaviours"andactualphysicalsystemormathematicalinterpretation. ThemostformaloriginsofUniversalitycomefromMitchellFeigenbaum[ 29 51 52 ], circa1975whenhediscoveredtheuniversalconstant =4 : 669 ::: .Originally was observedwithinasystemasthe\rate"ofonsetofperioddoublingas: 2 n +1 )]TJ/F2 1 Tf0.99999 0 Td[( 2 n 2 n +2 )]TJ/F2 1 Tf0.99999 0 Td[( 2 n +1 ,as n !1 ; (1{4) where 2 k isthe valueofonsetofthe2 k th cycle,intermsofFigure 1-12 itisthelimit ofthesequence b 1 b 2 b 2 b 3 b 3 b 4 ,... .Almostimmediatelythoughwiththisinitialobservation, cameboththetheoreticalandexperimentalconrmation(seeTable 1-3 )thatextendedthe existenceof fromwithinasinglesystemtoeverysystemthatundergoesperioddoubling. Moreimportantlythough,andtheactualbasisofUniversality,outofthetheoretical treatmentemergesa\convergentfunction"thatencapsulatesallsuchsystems.Avery rudimentaryapproachtodescribethisfunctionisthefollowing,consideranyfunctionthat relates\points"inasetinthemannerwehavebeenconcernedwithinthischapter. Further,considereachsuchfunctionasitselfa\point"inasetoffunctions,these \function-points"convergetoasingle\function-point",insomesensejustasthe b i b i +1 convergeto .Wehaveseenintheprevioussectionhow x n of F ,canallbevariedin somewaytocreateacombinationof F andanintervalfromwhichchaosemerges,nowwe seethatthefunctionalformof F itselfcanalsovary. 1.2.3ThresholdControlandExcessOverowPropagation Inthisshortsectionwewillintroducetwoprocesseswhichhaveextendedthe breadthofinuenceofchaostheory,andarewidelyusedinalgorithmsofchaosbased 43

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Figure1-12.Logisticmapbifurcationdiagramforsomevalueswithin3 << 1 .\//" ontheaxisindicatediscontinuityinthedisplayedpoints, b 1 and b 2 are actuallylongerthanshown.FortheLogisticmapthesequenceoftheratios b 1 b 2 b 2 b 3 b 3 b 4 ...convergesto .Inothersystemsthe b i canbedierentfromthe Logisticmap,buttheconvergentvalueoftheratiosisthesame.Inset: Figure 1-1 Table1-3.ExperimentalmeasurementsofFeigenbaum`sconstant( )indierentsystems basedontheirperioddoubling.(Adaptedfrom[ 53 ].) Systems Hydro dynamic Electronic Optic Acoustic W ater MercuryDio deT ransistor Laser Helium Observed 4 : 3 0 : 84 : 4 0 : 14 : 3 0 : 14 : 7 0 : 34 : 3 0 : 34 : 8 0 : 6 computation.Wewillnotgoindetailsforeitherprocess,asbotharenowquiteextensive intheirownright. Controlandsynchronisationofchaoticsystems[ 13 { 15 54 { 64 ]havebeenstudied foralmost20yearsnowwithnumerousresults;wesimplydemonstratethethreshold controlmechanism,asalreadymentionedinrelationtothesecondpartoftheSharkovsky theorem.Theotherprocessweintroduce,theexcessoverowpropagation[ 65 { 68 ],also 44

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hasalonghistoryandconnectionstoawidevarietyofothereldsascriticalphenomena, phasetransitions,cooperativebehaviours,andmore. ThresholdControlMechanism .FollowingSharkovsky'stheoremandspecically thesecondpart,weknowthatamapexistsforeveryperiodicityandmore,thatonce periodicity q isestablishedperiodicity p isguaranteed,where p C q .Thereforeifwestart withamapwhichhasperiod3,theLogisticmap F 4 forexample,thissamemapcanbe ofsomeother\maximal" 20 periodsimplybyconningthedomainofthemap,evenmore thisperiodwillseemattractive 21 .Basicallyinsteadoflookingforawholenewdierent mapof\maximal"periodicity q ,wetake F 4 : I I andconsiderthemapwhichisthe partof F 4 : J J ,where J =(0 ;x ]and x isthemaximalpointvalueofthesequenceof pointsthatarethedesiredorbit,hencethenameofthemethod\thresholdcontrol".The mechanismcanbedenedas: ~ F ( x )= 8 > > < > > : F ( x ),for F ( x ) x x ,for x
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Figure1-13.ThresholdControlMechanism. F 4 thedottedparabola,isconnedonthe interval J =(0 ;x ]andasaresultthemap ~ F 4 isproduced,thesolid parabola.Inthiscase x =0 : 915.Emptycircles, # ,markthreedierent initialconditionsandtheirpaths,indottedlines,showinghowtheyare \pushed"ontotheperiod4sequenceofpoints x 1 ;x 2 ;x 3 ;x 4 ,markedwithfull circles, ExcessOverowPropagation .Uptonowwehavebeenconsidering,insome formorother,justasinglechaoticsystem,andeventhoughtthepotentialofevena singlechaoticsystemisimmense,twoisalwaysmorethanone.Sinceweareconsidering morethanonesystemweneedtodenesomewayinwhichthetwosystemstointeract. Currentlythemethodusedinchaoticcomputationistheexcessoverowpropagation method.Wedenea\monitoringvalue", x ,forthestateofthe\emitting"system, f ( 1 x ); 46

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Figure1-14.Thresholdvaluesforconningthelogisticmaponorbitsofperiodicity2to 50. oncethis\monitoringvalue"iscrossedoverbytheactualstateofthesystem, f ( 1 x ) > x thedierencebetweentheactualstateofthesystemandthe\monitoringvalue",the excessoverow( f ( 1 x ) )]TJ/F4 1 Tf7.9706 0 0 7.9706 0 0 Tm24.735 47.944 Td[( x = E ),ispropagatedtothereceivingsystem(s) f ( 2 x + E ). Notethatthe\emitting"systemisnotaectedinanyway,unlikewiththethreshold controlmethodwhereweconnethestateofthesystem,alsoatthe\receiving"system theincorporationoftheexcessoverowcanhappenbothbeforeandafteraniterationof thesystem f ( 2 x )+ E .Themethodisverysimilartothethresholdcontrolmethod,but independent,inthesensethatthe\monitoringvalue"canactasathresholdcontrol,but isnotnecessary,thetwovaluescanbedierent,providingusmoreexibility. 1.3Conclusion Wehavepresented,asbrieyaspossible,partsofChaosTheorythatarethebasis forChaoticComputation.EventhoughthisexpositionwasmadeinthespiritofSet TheoryandLogic,theexactconnectionswerenotactuallylaidoutsinceourforthcoming 47

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presentationofalgorithms,developedforChaoticComputation,donotactuallyreachsuch lengths.Thepossibilityistherethrough,andweplantoexplorefurtherthisviewinour futurework. 48

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CHAPTER2 INTRODUCTIONTOCHAOTICCOMPUTATION Chaoticcomputationistheexploitationoftherichnessinbehavioursofchaotic systemstoperformcomputationaltasks.Thischapterdealswithdierentmanipulations ofstatevariablesthatleadtoselectionofaspecicbehaviour,withoutrelinquishing accesstootherbehaviours,demonstratingtheinherentversatility.Naturalparallelism emergesboththroughcooperativeprocessesofindependentchaoticsystemsandthrough exploitationofmulti-dimensionalsystems.Averygoodabstractionofchaoticcomputation isthetranslationoffunctionsanddataoperatorsfromthesoftwarerealmontothe hardware;directimplementationoftheobjectivetaskinhardware.Herespecicallywe almostexclusivelypresent,thetheoreticaldevelopmentofalgorithmsandmethodsof chaoticcomputaionandusetheLogisticmaptoillustratespecicexamples.Foractual physicalrealizations,andverications,oftheseresultswedirectthereadertoresultsof electronicimplementations,usingChua`scircuit[ 69 ],andLogisticmapcircuit[ 70 71 ], andtosimulationresults[ 26 ]ofaIRNH 3 laserusingtheLorentzsystem[ 72 73 ].In additionwecandirectthereadertotheveryrecentphysicalrealizationresultsforchaotic computationusingsynchronization[ 74 ]andstochasticresonance[ 75 ]. 2.1NumberEncoding Theprimaryrequirementforanycomputationalsystemisrepresentationofdata. Thereneedtobemethodsthroughwhichdatacanberecognized,stored,andreproduced, obviouslyonlythesethreeprocessesdonotmakeaverycapablecomputer.Theabsolute universallanguageismathematicsandsonumbersisthemostgenericformofdata representation.Thereforewebeginwiththedierentmethodsavailableinchaotic computationfornumberrepresentation. 2.1.1ExcessOverowasaNumber Thisisthesimplestmethodforencodingnumbersusingachaoticcomputerand followsfromtheexcessoverowpropagationmechanism.Foranygiventhreshold( x ),the 49

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amountbywhichthesystemvariablemonitoredexceedsthethreshold, E = f n ( x ) )]TJ/F2 1 Tf1.0246 0 Td[(x forsome n th iterationsuchthat f n ( x ) >x ,isnamed\excessoverow"andisusedto representaninteger.Morespecically,weutilizeaninterval, K 1 ,forwhichthedynamical systemthatdrivesourchaoticcomputercontainsxedpointsforasingleiteration, n =1, underthresholdcontrol,i.e. f ( x ) >x ; 8 x 2 K 1 .Givenarequirementasofrepresenting thesetofintegernumbers f 1 ;N g ,wecanndfromtheinterval K 1 the x whichproduces thelargestvalueofexcess E max and\equate"thiswiththelargestintegerwewishto encode, N .Asaresult,wecandene1 E max =N ,where iscalled\unitoverow" andeveryintegerintheset f 1 ;N g canberepresentedbyanexcessoverowof z ,where z 2f 1 ;N g .Obviouslytheinteger0isrepresentedbyzeroamountofexcess,obtainedby settingthesystematanaturalxedpoint,i.e. f n ( x )= x; 8 n AsanillustrativeexampleweusetheLogisticmap,Equation 1{1 .FortheLogistic maptheinterval[0 ; 0 : 75]producesxedpointsunderthresholdcontrol,notethat 0and0 : 75arethe\natural"xedpointsofthissystemandeithercanbeusedto representtheinteger0.Bytakingthederivativeof F ( x ) )]TJ/F2 1 Tf1.0782 0 Td[(x wendthethreshold thatproducesthemaximumexcessoverowat x =3 = 8,withanemittedexcessof E max =9 = 16,seeFigure 2-1 .Giventhesetofintegers[0 ; 100]wecanencodethem usingthresholdcontrolledlogisticmapelementsfollowingthestepsdescribedabove,i.e. 100 E max ) =(9 = 16) = 100and x = )]TJ/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm36.695 42.009 Td[(3+ p 3 2 )]TJ/F5 1 Tf0.8264 0 Td[(4 ( )]TJ/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm43.206 42.009 Td[(4) ( )]TJ/F3 1 Tf7.9706 0 0 7.9706 0 0 Tm45.643 42.009 Td[(z ) 2 ( )]TJ/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm42.193 40.593 Td[(4) ; 8 z 2 [0 ; 100],seeFigure 2-2 Theextensiveversatilityofthismethodwillbedemonstratedinfollowingsections. Wewillshowimplementationsofthismethodnotonlyforothernumberrepresentation methods,butalsoinalgorithmsfordecimalandbinaryarithmetic,aswellasforboolean logicoperations. 2.1.2PeriodicOrbitsforNumberRepresentation Themostimmediateextensiontotheexcessoverowencodingmethodistoconsider thebehaviourofthedynamicalsystemoutsidetheregionalreadybeingutilized,i.e. x= 2 K 1 .Wecanutilizetheeectofthethresholdcontrolmechanismonthisinterval, K 2 50

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Figure2-1.Emittedexcessbythresholdingthelogisticmapintheinterval[0 ; 0 : 75].The threshold x =3 = 8producesthelargestexcessof E max =9 = 16. tostabilizethesystemtoaperiodicorbit.FollowingSharkovsky'stheorem,seeTheorem 1.1 onpage 30 ,sincewehaveaperiodthreeorbit,orbitswithperiodsofallotherinteger values,andmore,areguaranteedtoexist.Thereforeitisanobviousextensiontoutilize theappropriateorbittorepresentitsrespectiveintegernumber.Morespecically,foreach n> 1wenda x 2 K 2 ,forwhich f n ( x ) >x and f m ( x )
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Figure2-2.Encodingthesetofintegers f 0 ; 1 ;:::; 100 g .Weuse100 E max =9 = 16to obtaineachexcess z andthethreshold x thatproducesit. overowmethodinathirdmethodforrepresentingnumbersmoreinthespiritofbinary representation. 2.1.3RepresentationofNumbersinBinary Inthisnalexampleofnumberencodingmethodswewillcombinetheprevioustwo methods,weusebothperiodicorbitsandexcessoverow.Wecanrepresentanumber inbinaryformatbycouplingtogetherelementswhichhavetheirperiodicitydetermined bytheirpositionawayfromtheradixpointas2 # digits )]TJ/F3 1 Tf0.8264 0 Td[(position .Thefartherawayfrom theradixpointanelementistheshorterperiodicitywegiveit,withthebitfarthest awaybeingonperiodone.Theelementsarejoinedtogetherserially,sothatthe\generic overow"generatedbyeachelementcascadesthroughthearrayuntilitreachestheopen endatthemostsignicantdigit,wherewehavethereadout,seeFigure 2-3 .Tobemore specicabinarynumber a N :::a 1 willberepresentedby N elements.Eachelement, j asintheencodingbasedonperiodicity,willbesetatathresholdsothattoproducea 52

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\genericoverow"atperiodicityof2 N )]TJ/F3 1 Tf0.8264 0 Td[(j ,ifthebinarydigit a j =1,andatathreshold of0,if a j =0,thusgeneratingnooverow.Theresultingarrayisupdated2 N )]TJ/F5 1 Tf0.8264 0 Td[(1 times resultinginamultipleoftheunitoverow P N j =1 a j 2 j )]TJ/F5 1 Tf0.8264 0 Td[(1 atthereadout. AnillustrativeexampleusingtheLogisticmapisshowninFigure 2-3 ,whereweare usingfourelements, N =4,torepresentthebinarynumber1111.Followingtheexample, a 4 ,thefarthestelementfromtheradixpoint,issettoathresholdthatemitsexcesson everyupdate,i.e.periodicityone( x 2 (0 ; 0 : 75)); a 3 isgivenathreshold x 2 (0 : 75 ; 0 : 905), toproduceexcessoneverysecondupdate; a 2 from x 2 (0 : 905 ; 0 : 925)foranexcessevery fourthupdate;and a 1 from x 2 (0 : 925 ; 0 : 926)foranexcesseveryeighthupdate.Asa resultwewillcollectemittedexcess8+4+2+1=15,respectivelyfromeachelement,as isdesiredforencoding1111inbinary. Thismethodallowsustoencodelargenumberswithouttheneedofthresholdsthat produceproportionallylargeperiodicities,orofthresholdsthatproduceexcessinsteps ofverysmall .Thisisagoodexampleoftheexibilityofchaoticcomputation.Inthe specicexamplewe\sacriced"eciency,inthenumberofelementsperencodedinteger, toallowshortperiodicitiestoencodelargernumbers.Obviously,themethodcanbeeasily modiedtoencodenumbersinanyotherbaserepresentation. 2.2ArithmeticOperations Wehaveestablishednotone,butthreemethodsforrepresentingnumbers,weshould seetheminaction.Themostobviousstartingpointisthesimplestarithmeticoperation: addition,whichweextendintomultiplication,whichweextendtotheleastcommon multipleproblem. 2.2.1DecimalAddition Therearemultiplechaoticcomputationalgorithmsfordecimaladdition.Wewill presentthreesuchalgorithms,eachonebuildingonitspredecessor,fromserialadditionto paralleltobranching. 53

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Figure2-3.Numberencodinginbinaryformat.Theexcessfromeachelementcascadesto theone\above"ituntilthereadoutisreached.Theelementclosesttothe readout, a 4 ,emitsoneveryupdate,andaswemove\down"thechainthe elementsemitinincreasingpowersof2.Theoverallresultisafter8updates, wehave8unitsofexcessfrom a 4 ,4from a 3 ,2from a 2 and1from a 1 ,giving usatotalof15units.Anybinarynumbercanberepresentedwiththis method.(Adaptedfrom[ 26 ].) Serial .Themoststraightforwardalgorithmforadditionutilizestheexcessoverow encodingmethodinaserialmanner.Inrealityitisanaturalextensionoftheencoding method,sinceeachnumberisencodedasaproportionalexcesswecanchain-linkthe elementsandcascadetheemittedexcessfromeachelementintoitsneighbourallthe waytotheendofthechain.Eachexcess\buildsup"ontheoneafterit,all\naturally" summingupattheedgeofthechain,seeFigure 2-4 .AswehaveseeninSection 2.1.1 the\unitoverow"( )worksastheproportionalityconstantbetweenintegersand emittedexcess,thereforetheadditionofintegersi,j,k,lissimplyreplicatedby 54

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i + j + k + l =( i + j + k + l ) ,usingtheavalanchingoftheexcesses.For thisalgorithmthecomputationaltimeisdependentontheadaptiveprocessandthe numberoftermsinthesum.Specicallyafterasinglechaoticupdateofalltheelements, ittakesasmanyadaptationstepsastherearetermsinthesumtocompletetheoperation. Figure2-4.SerialAddition.Werecruitasmanyelementsastermsinagivensum.Each elementisassignedanumberfromthesum,encodedusingtheexcessoverow method.Theelementsarecoupledtogetherinachainsuchthattheexcesscan owdownthechain.Theresultisattheopenendwecollectthesumofall theemittedexcessesasamultipleof .(Adaptedfrom[ 25 ].) ParallelandBranching .Followingfromthepreviousexample,considerthesum offourterms,i,j,k,lusingtheserialadditionalgorithm.Ifwetakeacloserlookat thedynamicsofthe\last"elementinthechain(inthiscasetheoneencodingl),wesee thatthiselementwillreceivetheexcessofallprecedingelements\simultaneously".To visualizethisconsiderthe\local"dynamicsofthreeelements,oncethechaoticupdate iscomplete.Theexcessoftherstelementisavalanchedtothesecond,whereit\builds up"withthelocalexcessatthesecondelement,sothethirdelementwillreceivethe combinedexcessofthetwoprevioustoitasasingleexcess.Thiscanalsobeachievedby introducingthetwoexcessesindependentlyofeachother,butsimultaneously,i.e.arrive 55

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atthesametimeand\buildup"oneachotherlocallyatthethirdelementinsteadofat thesecond.Ofcoursethiscanbeextendedtoanynumberofelementsprecedingthe\last" elementinthechain.Turningbacktoourexampleofaddingi,j,k,l,thetopologyofthe connectivityinsteadofbeingachainisnowatreediagram,seeFigure 2-5 .Byturning thechainintoatreewehavecollapsedtheserialadditionalgorithmtoatwostepserial addition,regardlessofthenumberoftermsinthesum.Therststepistosumall,but oneofthetermsinparallelandthenseriallycombinetheparallelsumtotheremaining term,beforereadingtheresultattheopenend.Weneedtonotethatwecannotfully parallelizetheoperationbyconnectingalltermsdirectlytotheopenend,sincewearenot attributinganydynamicalpropertiestotheopenend,i.e.iftheopenendhadtheability tocorrectly\buildup"excessesitwouldbeidenticaltoalltheotherdynamicalsystems usedinthesum,makingitthe\last"element 1 Obviouslyforasumof N termstheshortestcomputationaltimefortheoperationis toconnect N )]TJ/F0 1 Tf1.0454 0 Td[(1termsinparalleltothe\last"termandperformtheoperationintwo steps,thiswouldbeanalogoustoincreasingthenumberof\branches"inthenetwork.In casethoughofconnectivityand/orspatialrestrictionswecouldalsoextendthealgorithm byincreasingthenumberof\trunks"inthenetwork,asisshowninFigure 2-6 ;notethat nowourcomputationaltimeisdependantonthenumberof\trunks".Thisisanothercase ofchaoticcomputationexhibitingitsexibility,wecan\sacrice"temporalperformance tosatisfyspatialconstraints. 2.2.2BinaryAddition Extendingtheaboveadditionalgorithmsforthebinarynumberencodingmethod isstraightforward.Theserialadditionofbinarynumbersisrealizedbyconnectingthe \endbit"ofonenumberwiththe\endbit"ofthefollowingnumberinachainmanner allthewaytothelastnumberandthentotheopenend,seeFigure 2-7 .Similarlyfor 1 Thiscouldactuallybeovercomebyinsertingtoanysum,0asthelastterm. 56

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Figure2-5.Decimalparalleladdition.Theexcessfromtheelementsencodingi,j,kis simultaneouslypropagatedtotheelementencodinglonarstavalanching step,andonthesecondavalanchingstepthecollectivesumisreadattheopen edge.(Adaptedfrom[ 26 ].) Figure2-6.Thebranchingalgorithmcanbeextendedtoalargertreelikestructure.The computationaltimeinthiscaseisproportionaltothenumberofbranchesin thelongestpaththatterminatesatthe\last"element.Theabovenetworkwill sum15termsinfouravalanchingsteps.(Adaptedfrom[ 26 ].) 57

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implementingtheparalleladditionmethodforbinarynumbersweconnectthe\end bit"ofalltermsinthesum,butone,tothe\endbit"ofthesingletermchosentoact asthecollectionhubforalltheexcesses,beforesendingtheresulttotheopenend,see Figure 2-8 Figure2-7.Schematicrepresentationoftheserialadditionmethodforbinarynumbers. Thenumbers7,5,2,1areencodedasexplainedaboveinSection 2.1.2 andthe elementsareseriallyconnected.Theexcessoverowbuildsupasitmoves troughthenetworkand15unitsofexcessarecollectedattheOUTPUT. (Adaptedfrom[ 26 ].) 58

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Figure2-8.Schematicrepresentationoftheparalleladditionmethodforbinarynumbers. Inthiscaseabranchingtopologyisusedforthenetwork,whereoneofthe systemsactsasacollectionhubforsimultaneousbuildupofexcess.(Adapted from[ 26 ].) 2.2.3DecimalMultiplicationandLeastCommonMultiple Wecanextendanyadditionmethodtoperformmultiplicationintheusualway, theproductoftwonumbers m n isasumof n termsofthenumber m (andofcourse visaversa m termsofthenumber n ).Withourtwoaboveadditionmethodswehave twoobviouswaysofimplementingmultiplicationtheserialandparallelsummationof theterms.Furthermoreusingtheexcessoverowencodingofnumberswehaveathird method;wecanuseasingledynamicalsystemthatemitstheappropriateexcessonevery update,specicallytheamountthatrepresentsoneofthenumbers, m forinstance,and weupdatethesystem n timescollectingatotalofexcessequalto( m n ) ,theproduct 59

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ofthemultiplication.Thisthirdmethodutilizestimeasacomputationalquantity,and leadstoafourthmultiplicationmethod,whichwewillalsouseasasteppingstoneforthe methodofleastcommonmultipleofmanynumbers. Followingtheperiodicorbitencodingmethod,wecanrepresenteachnumber,of atwonumberproduct,usingforeachnumberachaoticelementsettoemitexcessat theappropriateperiodicity.Thereforegivenaproduct m n weutilizetwodynamical systems,onesettoemitevery m updatesandtheothertoemitevery n updates,the productofthetwonumbersisgivenbythenumberofupdates on whichthetwoelements emitsimultaneously;morespecicallytheelementemittingevery m updateswillhaveits n th emissiononthe( m n ) th updateandthesameappliesfortheotherelement.The simpleextensionofthismethodtoalargernumberoftermsresultsinanalgorithmforthe leastcommonmultipleofallthetermsinconsideration 2 2.3BinaryOperations Thepowerofconventionalcomputersliesintheirabilitytoperformbooleanalgebra. Evensotheirbuildingblocksarerestrictedbymanufacturetooneofthetwofundamental gates,NORorNAND,whichwithsuitablecombinations,ofeither,theotherlogicgates canbereproduced,forexampleAND(X,Y) NOR(NOR(X,X),NOR(Y,Y)).Thisclearly isnotthemostecientmethodforbooleanalgebra;amuchmoreecientalternative isforeachgatetorequireonlyonebuildingblock,converselyeachbuildingblocktobe abletoperformallgates.Weshowhowasinglechaoticelementcanrepresenteachofall logicgatesthroughsimplestatemanipulations,removingtheneedforcombiningelements [ 32 76 ].Furthermoreweshowhowmulti-dimensionalitycanleadtonaturalparallelism, andwegoevenfurther,exitingthecapabilitiesofconventionalalgorithmsandaddressing aproblemdesignedforquantumcomputation. 2 Thecaseofthreenumbersinaproductishandledserially,ndtheproductoftwoof theterms,andmultiplyitbythethirdterm. 60

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Figure2-9.SchematicrepresentationofthemethodforcomputingtheLeastCommon Multipleoffournumbers.Eachelementemitsexcesstothecollectingelement attheappropriateperiodicityforitsencodednumber.Theresultatthe OUTPUTisexcessofthecollectingelementwithmagnitudeequaltothe numberoftermelementsthatemittedonthecurrentupdate.Thenumberof updatesthatcausesthecollectingelementtoemitexcessequaltothenumber ofterms(i.e.alltermelementsemittingsimultaneously)istheLeastCommon Multipleoftheterms.(Adaptedfrom[ 26 ].) 2.3.1LogicGates Aswehaveshownanimportantcharacteristicofchaoticcomputationisthe versatilitywehaveinimplementingthesamealgorithmwithdierentmethods.This ofcourseextendstoimplementationsforrepresentationoflogicgates,i.e.thereare multiplewayswecanachievethisrepresentation.Inthissectionwefocusonthemost 61

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straightforwardmethoddevelopedandinordertopresentthemethodmoreclearlywe specifyittothelogisticmap 3 ,Equation 1{1 Themethodweuseconsistsofthreesteps:(a)initialization,(b)chaoticupdate,and (c)thresholdcontrolandexcessoverow.Comparedtothemethodspresentedinprevious sections,thenewconceptforthismethodisinitialization;i.e.thesettingoftheinitial stateofthesystem,( X 0 )justbeforetherstchaoticupdate,basedonspecicrules.This initialconditionofanelementisusedtodenewhichlogicoperationitperformsandon whatsetofinputs.Specically,weinitializealogisticmapelementbysettingitsinitial value x 0 accordingto: x 0 = x prog + x I 1 + x I 2 ,forgatesthatoperateontwoinputs, x 0 = x prog + x I ,forgatesthatoperateononeinput, x prog canbethoughtofas\programming"thegateand x I i astheinputvalues.Foran inputoflogical1, x I i = ,andforaninputoflogical0, x Ii =0.Asbeforeachaotic updateimpliestheapplicationofthelogisticmap: x 0 F ( x 0 ).Thecontrolandoverow mechanismremainsthesameaswell: E =0if F ( x 0 ) x ,and E = F ( x 0 ) )]TJ/F2 1 Tf1.0939 0 Td[(x if F ( x 0 ) >x .Where x isthethresholdfortheelementand E theexcessoverow generated.Hereinthecontextofbinaryalgebrawherethesetofintegerscontainsonly0 and1, E and areactuallyequivalent 4 Turningtothespeciclogicgatestobeimplemented,thefollowingTable 2-1 summarizestheinput-outputrelationshipswearetorepresent. Thetaskistoidentifyinitialconditions, x prog + x I i ,andthresholdvalues, x ,for whichachaoticupdatewillresultin F ( x 0 ) x forwheretheoutputsintheabovetable 3 Universalityofchaoticsystemsallowsustoassumedemonstrationsonthelogistic mapcanbecarriedovertoanyotherchaoticsystem. 4 Equivalenceofinputsandoutputsisactuallya\soft"requirement. 62

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Table2-1.Truth-tableforthelogicoperationsAND,OR,XOR,NOR,NAND,NOT,and theidentityoperation (WIRE). I 1 I 2 ANDORXORNOR NAND INOT WIRE 0 00001 1 01 0 0 10110 1 10 1 1 00110 1 1 11100 0 are0,and F ( x 0 ) >x wheretheoutputis1.Asaspecicexample,fortheORgatewe havethefollowingrequirements: 1. I 1 = I 2 =0,whichimplies x I 1 = x I 2 =0,i.e. x 0 = x prog .Therequiredoutputis0, whichimplies F ( x prog ) x 2. I 1 =0and I 2 =1,whichimplies x I 1 =0and x I 2 = ,i.e. x 0 = x prog + .Therequired outputis1,whichimplies F ( x prog + ) )]TJ/F2 1 Tf1.0253 0 Td[(x = .Thisrequirementissymmetricto I 1 =1and I 2 =0,sotheconditionsforsatisfyingbothrequirementsareidentical. 3. I 1 = I 2 =1,whichimplies x I 1 = x I 2 = ,i.e. x 0 = x prog +2 .Therequiredoutput is1,whichimplies F ( x prog +2 ) )]TJ/F2 1 Tf0.99999 0 Td[(x = Alloftheaboverequirementsneedtobesatisedbythesamevaluesfor x prog and x suchthatallthreeconditionsholdtrueregardlessofinputs.Inasimilarmannerwecan providerequiredconditionsforachaoticelementtorepresenteverygate,seeTable 2-2 Table2-2.Necessaryandsucientconditionsforachaoticelementtosatisfythelogic operationsAND,OR,XOR,NOR,NAND,NOT,andtheidentityoperation (WIRE). Input x I 1 + x I 2 ANDORX OR 0 F ( x prog ) x F ( x prog ) x F ( x prog ) x F ( x prog + ) x F ( x prog + ) )]TJ/F2 1 Tf0.99999 0 Td[(x F ( x prog + ) )]TJ/F2 1 Tf0.99999 0 Td[(x 2 F ( x prog +2 ) )]TJ/F2 1 Tf0.99999 0 Td[(x F ( x prog +2 ) )]TJ/F2 1 Tf0.99998 0 Td[(x F ( x prog +2 ) x Input x I 1 + x I 2 NOR NAND 0 F ( x prog ) )]TJ/F2 1 Tf0.99999 0 Td[(x F ( x prog ) )]TJ/F2 1 Tf0.99999 0 Td[(x F ( x prog + ) x F ( x prog + ) )]TJ/F2 1 Tf0.99999 0 Td[(x 2 F ( x prog +2 ) x F ( x prog +2 ) x Input x I NOT WIRE 0 F ( x prog ) )]TJ/F2 1 Tf0.99999 0 Td[(x F ( x prog ) x F ( x prog + ) x F ( x prog + ) )]TJ/F2 1 Tf0.99999 0 Td[(x 63

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Valuesthatsimultaneouslysatisfytheaboveconditionsareeasilyfound.Specically bychoosing =0 : 25,theORgatecanberealizedbychoosingvaluesfor x prog =1 = 8and for x =11 = 16: 1. F ( x prog )= F (1 = 8)=7 = 16 x (=11 = 16), 2. F ( x prog + )= F (3 = 8) )]TJ/F0 1 Tf0.99999 0 Td[(11 = 16=15 = 16 )]TJ/F0 1 Tf0.99999 0 Td[(11 = 16=1 = 4(= ), 3. F ( x prog +2 )= F (5 = 8) )]TJ/F0 1 Tf0.99999 0 Td[(11 = 16=15 = 16 )]TJ/F0 1 Tf0.99999 0 Td[(11 = 16=1 = 4(= ). Infactvaluesthatsatisfytheconditionsforallthegatesand =0 : 25,havebeen identiedandaresummarizedinTable 2-3 Table2-3.Initialvalues, x prog ,andthresholdvalues, x ,requiredtoimplementthelogic gatesAND,OR,XOR,NOR,NAND,NOT,andtheidentityoperation (WIRE),with =0 : 25. V alueANDORXORNORNANDNOT WIRE x pr og 01 = 81 = 41 = 23 = 81 = 21 = 4 x 3 = 411 = 163 = 43 = 411 = 163 = 43 = 4 Thefactthatwehaveamethodtoimplementalllogicgatesisnotimpressive,whatis impressiveisthefactthatwecan switch fromonegatetoanotherfromonecomputational steptothenext,andevenmore,switchveryeasilyand\fast','inarelativetimescale. Thisleadstotheconceptofon-the-yhardwarere-programming;anarchitecturebasedon conventionalcomputation,soallothercomponentscanbeeasilyimported,butwiththe eciencychaoticcomputationoers. 2.3.2ParallelLogicandtheHalfAdder Inthissectionwepresentwhatisprobablythemostimportantextensionof performingbooleanlogicwithchaoticsystems[ 32 ].Thesameprocedureweillustrated aboveforthelogisticmapisimplementedonthe2-dimensionalneuronmodel[ 77 ]: x n =( x n )]TJ/F5 1 Tf0.8264 0 Td[(1 ) 2 exp( y n )]TJ/F5 1 Tf0.8264 0 Td[(1 )]TJ/F2 1 Tf0.99999 0 Td[(x n )]TJ/F5 1 Tf0.8264 0 Td[(1 )+ k (2{1) y n = a y n )]TJ/F5 1 Tf0.8264 0 Td[(1 )]TJ/F2 1 Tf0.99999 0 Td[(b x n )]TJ/F5 1 Tf0.8264 0 Td[(1 + c (2{2) 64

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where a =0 : 89 ;b =0 : 18 ;c =0 : 28 ;k =0 : 03;theseparametervalueskeepthemodel dynamicscompletelychaotic. Twodistinctcasesareinvestigated: 1.ThepossibilityofperformingXORgateandANDgateinparallel(HalfAdder) 2.PerformingtwoANDgatesindependently. Therstcasespecically,istheapplicationofthetwogatesonthesamesetofinputs, eachgatebeingperformedinadierentdimension.Thesecondcaseinvolvesoperatingthe twoANDgatesondierentsetsofinputs,againeachdimensionperformingoneoperation. Aswiththecaseofthelogisticmap,therststepistodenethenecessaryconditions neededtosatisfythetruthtableofeachcase,specicallyfollowingthetruthtables (Tables 2-4 2-5 )weconvertthemtotheconditionalTables 2-6 2-7 .Followingthe processoftheprevioussectionwedeterminevaluesforthe\programming"stateshiftand thresholdvaluethatsatisfytheseconditions,seeTables 2-8 2-9 Table2-4.TruthtableforXORandANDlogicgatesonthesamesetofinputs.(Case 1 ) I 1 I 2 XOR AND 0 00 0 0 11 0 1 01 0 1 10 1 Weshouldmentionthateachcaseisinvestigatedindependently,i.e.thereisno requirementthatthenumberofiterations( n )isidenticalforbothcases,orthatthevalue representingalogical1( )isthesame. Wehavealreadyseensometypeofparalleloperationswithchaoticcomputing inSection 2.2.1 withtheparalleladdition,herethoughweseeacleardemonstration ofparallelism.Thedynamicalsystemperformstwocompletelydierentoperations simultaneously.Infactthenextsectionbuildsfurtherontheparallelcapabilitiesof chaoticcomputation,bytacklingthecomplexDeutsch-Joszaproblem. 65

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Table2-5.TruthtablefortwoANDgatesoperatingonindependentinputs.(Case 2 ) I 1 1 I 1 2 I 2 1 I 2 2 AND( I 1 1 ;I 1 2 )AND( I 2 1 ;I 2 2 ) 0 00 0 0 0 0 00 1 0 0 0 01 0 0 0 0 01 1 0 1 0 10 0 0 0 0 10 1 0 0 0 11 0 0 0 0 11 1 0 1 1 00 0 0 0 1 00 1 0 0 1 01 0 0 0 1 01 1 0 1 1 10 0 1 0 1 10 1 1 0 1 11 0 1 0 1 11 1 1 1 Table2-6.RequiredconditionstosatisedparallelimplementationoftheXORandAND gate.(Case 1 ) Initial conditionsXOR AND x prog ;y pr og x n x y n y x prog + 1 ;y prog + 2 x n )]TJ/F2 1 Tf0.99999 0 Td[(x 1 y n y x prog +2 1 ;y prog +2 2 x n x y n )]TJ/F2 1 Tf0.99999 0 Td[(y 2 Table2-7.RequiredconditionsforimplementingtwoANDgatesonindependentsetsof inputs.(Case 2 ) Initial conditionsAND( I 1 1 ;I 1 2 )AND( I 2 1 ;I 2 2 ) x prog ;y pr og x n x y n y x prog ;y prog + 2 x n x y n y x prog ;y prog +2 2 x n x y n )]TJ/F2 1 Tf0.99999 0 Td[(y 2 x prog + 1 ;y pr og x n x y n y x prog + 1 ;y prog + 2 x n x y n y x prog + 1 ;y prog +2 2 x n x y n )]TJ/F2 1 Tf0.99999 0 Td[(y 2 x prog +2 1 ;y pr og x n )]TJ/F2 1 Tf0.99999 0 Td[(x 1 y n y x prog +2 1 ;y prog + 2 x n )]TJ/F2 1 Tf0.99999 0 Td[(x 1 y n y x prog +2 1 ;y prog +2 2 x n )]TJ/F2 1 Tf0.99999 0 Td[(x 1 y n )]TJ/F2 1 Tf0.99999 0 Td[(y 2 66

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Table2-8.Examplesofinitialvalues, x prog ;y prog ,andthresholds x ;y ,thatsatisfythe conditionspresentedinTable 2-6 ,yieldingtheparalleloperationofXORand ANDgates.Inthisrange 0 : 7,andthenumberofrequirediterationsto yieldthecorrectresultis n =10.(Case 1 ) x prog y pr og x y 1.8 1.70.441.11 1.4 1.650.38 1.07 Table2-9.Examplesofinitialvalues x prog ;y prog ,andthresholds,thatsatisfythe conditionspresentedinTable 2-7 ,yieldingoperationoftwoANDgateson independentinputs.Forthesevalues, 0 : 0115andthenumberofiterations required n =20;notethatthehighnumberofrequirediterationsandtheneed formoredecimalprecisionareadirectconsequenceofthelargenumberof requiredconditionstobesatisedsimultaneously.(Case 2 ) x prog y pr og x y 0.5375 1.08350.231 1.454 Animportantdierencefromtheprevioussectiononsinglebinarygatesisthatin thismethodthenumberofiterationsthatarenecessaryfortheconditionstobesatised isgreaterthan1,actuallyinChapter 5 wereportextensiveprogressininvolvingthetime dimensionwithchaoticcomputation. 2.3.3TheDeutsch-JozsaProblem TheDeutsch-Jozsaproblemanditssolutionalgorithm[ 78 ]isacelebratedresultof quantumcomputing,sinceitistherstexampleofquantumcomputingoutperforming classicalalgorithms.Inadditionithasbeenthesteppingstonefortheothertwo majorresultsofquantumcomputing,Shor'sfactoringalgorithm[ 79 ]andGrover's searchalgorithm[ 80 ](detailsofthechaoticcomputationsearchalgorithmaregivenin Chapter 3 ). Inthissectionwewilldemonstratethechaoticcomputingalgorithmforsolvingthe Deutsch-Jozsaproblem,whichisjustasecientasitsquantumcounterpart,provides theanswerinasinglecomputationalstep,butincontrastismuchmorerealizableand providesamoreapparentresult. 67

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Theproblemcanbestatedasfollows:Givenabinarydomain,i.e.adiscretedomain of2 k states,andgivenanarbitrarybinaryfunction f : f 0 ; 1 g k !f 0 ; 1 g ,i.e.afunction thatmapseverypointinthebinarydomaintoa0ora1,determinewhetherthefunction isconstant,mapsthewholedomainoneither0or1exclusively,orwhetherthefunctionis balanced,mapsthewholedomaininequaltermson0and1 5 Inotherwords,evaluate f foreverygivenpoint 6 ,andcountthenumberofresulting 0sand1sforallpoints,whichbasicallyistheconventionalapproachinsolvingthe problem.Asaresulttheconventionalalgorithminthebestcaserequiresevaluating thefunctionforonly2points,i.e.thefunctionresultsina0(or1)fortherstpoint considered,andtheoppositeforthesecondpointthusthefunctionisbalanced.Inthe worstcasethough,i.e.whentherstN/2pointsgivethesameresult,itwouldtake 2 k )]TJ/F5 1 Tf0.8264 0 Td[(1 +1evaluationstoconcludewhetherthefunctionisconstantorbalanced,i.e.the numberofcomputationalstepsneededgrowsexponentiallywiththenumberofbitsthat denethedomain. Inthecontextofchaoticcomputationthesituationismuchsimplersincewecan performthegivenfunctiononalldomainpointssimultaneously,furthermorewecanread theresultinonestep.Tomakethedemonstrationofthealgorithmclearerweexplicitly usethetentmap,Equation 1{3 andworkinastatespacethatappliestothismap.Also wepartitiontheexplanationofthealgorithminthreesteps:(a)denitionofthedomain, (b)denitionofthefunctionspace,and(c)implementationofthefunctionalongwith readingtheresult.Asistypicalofchaoticcomputationthereareatleastthreedierent 5 Weareguaranteedbytheproblemthatthefunctionwillbeeitherconstantor balanced,exclusively. 6 Thenumberofpointsinagivendomainisgivenas: N =2 k ,where k isthenumberof binarydigitsconsidered. 68

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implementationsofthealgorithm[ 26 ],belowwepresent,withsomemodicationsfrom [ 27 ],themostelaborate,butclearrealization. (a)Considerabinarydomainspace( B k )denedbypointsoflength k binarydigits, thereforethenumberofpointsin B k isequalto N =2 k .Sincewearetoworkwiththe tentmapwewilltranslateeverypointin B k ,ontoapointinthedomainofthetentmap [0 ; 1].Takeanyofthe2 k pointsas a 1 a 2 a 3 :::a k ,whereeach a i 2f 0 ; 1 g ,wemapeachsuch pointonto[0 ; 1]using 7 : x =2 )]TJ/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm34.212 73.007 Td[(( k +1) + k X i =1 a i 2 )]TJ/F3 1 Tf7.9706 0 0 7.9706 0 0 Tm43.979 73.007 Td[(i (2{3) Weencodethewholedomainonanarrayof N elements,denotedby X N ,each element( j )havingastate x ( j )givenbyEquation 2{3 .Toillustratethis,considerthe casewhere k =3,thebinarydomain( B 3 )haseightpoints f 000 ; 001 ; 010 ;:::; 111 g which translateontothetentmapdomainas f 1 16 ; 3 16 ; 5 16 ; 7 16 ; 9 16 ; 11 16 ; 13 16 ; 15 16 g ,eachofthesepoints isusedasthestatevalueofadynamicalelementinaneightelementarray.Sothewhole domaininconsiderationisencodedbyasinglearrayas: X 8 = f x (1)= 1 16 x (2)= 3 16 x (3)= 5 16 x (4)= 7 16 x (5)= 9 16 x (6)= 11 16 x (7)= 13 16 x (8)= 15 16 g (b)Thefunctionspaceofthefunctions: f : B k !f 0 ; 1 g consistsof2 2 k functions, thatisforeachofthe N =2 k pointstherearetwooutputpossibilities.Regardless ofthetotalnumberofpossiblefunctionsonlytwoareconstant,thefunctionthathas outputsall1andthefunctionthathasoutputsall0,forallinputpoints.Thenumber ofbalancedfunctionsthough,isdependanton k andisgivenbysimplecombinatoricsas C L M = L ( L )]TJ/F3 1 Tf0.8264 0 Td[(M )! M ,whereforasetof L items,madeupofonlytwodistinctobjects, C isthe numberofcombinationswherethereare M itemsofoneofthetwoobjects.Inthecase ofbalancedfunctionsforabinarydomainof N pointstheaboverelationshipbecomes 7 The2 )]TJ/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm16.215 19.158 Td[(( k +1) termisaddedsothatpoints0and0 : 5arenotusedforencoding. 69

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C N N= 2 = N ( N )]TJ/F5 1 Tf0.8264 0 Td[(( N= 2))!( N= 2)! .Morespecicallyforourexampleof k =3,wehaveatotalnumber ofpossiblefunctions2 2 3 =256,outofwhichtwoareconstantand( C 8 4 =)70arebalanced. Inourcase,ofthefunctiondomainbeingrepresentedbyanarrayoftentmapsas describedabove,thefunctionspacewillbepopulatedwithfunctionswhichareconstructed outofcombinationsofthefollowingtwobasisfunctions: x n +1 = T ( x n )=1 )]TJ/F0 1 Tf0.99999 0 Td[(2 x n )]TJET363.38 629.56 m369.29 629.56 lSBT/F0 1 Tf11.956 0 0 11.956 0 0 Tm30.396 53.081 Td[(1 2 (2{4) x n +1 = e T ( x n )=1 )]TJ/F2 1 Tf0.99998 0 Td[(T ( x n )=2 x n )]TJET409.03 596.88 m414.94 596.88 lSBT/F0 1 Tf11.956 0 0 11.956 0 0 Tm34.212 50.348 Td[(1 2 (2{5) where T isthetentmapand e T theinvertedtentmap,and n isusedtodenotethetime step.Eachoneofthe2 2 k functionsisconstructedasoneofthedierentcombinationsof thetwobasisfunctionsoflength2 k ,specicallythefunctionspacelookslike: f T (1) T (2) :::T ( j ) ;T (1) T (2) ::: e T ( j ) ;::: :::;T (1) e T (2) :::T ( j )]TJ/F0 1 Tf0.99999 0 Td[(1) e T ( j ) ;::: :::; e T (1) e T (2) ::: e T ( j ) g where j =2 k ,andeachsequenceoflength j ofthetwobasisfunctions,isoneofthe possiblefunctions( F )tobeappliedonthedomainspace 8 Referringbacktoourconcreteexampleof k =3,wehave256combinationsof T and e T inthefunctionspace,rangingfromthesinglesequenceofeightconsecutive T ,to70 combinationsoffour f : f 0 ; 1 g k !f 0 g andfour f : f 0 ; 1 g k !f 1 g ,tothesinglecaseof eightconsecutive e T ;andofcoursetheother184combinations 9 8 Inournotationhereweuse j insteadof N toindicatetherelationbetweenthe functionsandthearrayofdynamicalelementsencodingthedomainspace. 9 Carenottoconfusethe70functionsoffour T andfour e T withthe70balanced functions!Afunctionthatisbalancedin T and e T isnotnecessarilybalancedinits output. 70

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Figure2-10.Basisfunction T -TentMap. (c)Turningtothespecicprocessesinvolvedinsolvingtheproblem,wewillfocus ontheexampleof k =3tomakethissectionmoreillustrative.Aswehaveseenfor the3-bitcasethereare256functions,inthecontextoftheproblem,wearegivenone ofthesefunctions( F )(i.e.aspecicsequenceof8 T and/or e T )andareassuredthatit iseitherconstantorbalanced,thetaskistondwhichoneitis.Inastraightforward mannerwesetupthearrayofdynamicalelements, X 8 ,toencodethebinarydomain asexplainedin(a)andthenweapplythegivenfunctiononthewholearray F ( X n 8 )= X n +1 8 ,where n denotesthetimestep.Downtothescaleoftheindividualelements wehave T ( x n ( j ))= x n +1 ( j )and e T ( x n ( j ))= x n +1 ( j ),where j =1 ;:::; 8,whether 71

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Figure2-11.Basisfunction e T -InvertedTentMap. itis T or e T appliedonthe j th elementisbasedonwhichfunction( F )wearegiven. Oncethefunctionisappliedwethresholdtheelementsat x =0 : 5andcollectthe excesstheusualway.Basicallyweareusing0.5asaseparatrixofthestatespace,i.e. x n +1 ( j ) > 0 : 5 1 ) x n +1 ( j ) )]TJ/F0 1 Tf1.0359 0 Td[(0 : 5= E j and x n +1 ( j ) 0 : 5 0 ;E j =0(where E is emittedexcess),thisisastandardapplicationofsymbolicdynamics. Nowwehavethecollectedexcessinasinglestepwealsohavetheanswer:Ifthe collectedexcessis0,thenthefunctionwehaveistheconstantfunction F : B k !f 0 g k 72

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i.e.thefunctionthathasoutputall0;Ifthecollectedexcess 10 is2 k = 4=8 = 4=2,whichis themaximumpossibleexcessfor B 3 ,thenwehavetheconstantfunction F : B k !f 1 g k i.e.thefunctionthathasoutputall1.Obviouslyifourcollectedexcessissomeother valuewehaveabalancedfunction.Thisisasmuchaswearerequiredbytheoriginal problem,butwecandoevenbetter,usingtheexcesscollectedwecandetermineinwhich ofveclassesthegivenbalancedfunctionbelongsto. Theimportanceofthisalgorithmisnotintheactualtaskitaccomplishes,which isoflittlepracticaluse.Theprimaryimportanceis,likeforthequantumanalogue,in demonstratinganextremelyhighereciencythanconventionalcomputationinsolving problemsofthisclass.Furtherthoughforchaoticcomputationitisamilestoneagainst quantumcomputationaswell.Itshowshowchaoticcomputationcanhandleproblemsas wellasquantumcomputation,ifnotbetter. 2.4Conclusion Thischapteralmostexclusivelydealtwiththeoreticaldevelopmentsduringthe rstfouryearsofChaoticComputation(1998-2002),whilewerelegatedtheexperimental realizationstothereferences,sothattomaintainauniformglobaltonetothisdissertation. Inclosingandwithoutwishingtounderminetheotherresults,onceagainwedraw attentiontotheimportantresultofthesolutiontotheDeutsch-Jozsaproblem,andask thereadertoconsidertheconnectionsbetweenthisproblem,SetTheory,Logic,andtobe morespecictothestructureofachaoticfunctionattheFeigenbaumpoint. 10 Thefactorof 1 4 comesfromthefactthattheaverageexcessemitted h E j i is0.25. 73

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Figure2-12.FourrealizationsofthechaoticDeutsch-Jozsaalgorithmforthecase k =3. Thereareeightbinaryinputs000,001,010,011,100,101,110,111and throughEquation 2{3 weobtainthestatevalues 1 16 ; 3 16 ; 5 16 ; 7 16 ; 9 16 ; 11 16 ; 13 16 ; 15 16 eachgiventoanelementofanarray.Theverticallinesmarkthestatevalue ofeachelement x ( j ).Thehorizontallineistheseparatrixat0.5.(a)Given theconstantfunction F : B k !f 1 g k ,thisisthefunction: e T e TTTTT e T e T ,the applicationofthefunctionbringseveryelementover0.5,themaximum excessof2isemittedfromthearray.(b)Giventheconstantfunction F : B k !f 0 g k TT e T e T e T e TTT ,allarrayelementsremainunder0.5resultingin zeroexcess.(c)Giventhebalancedfunctionofeight e T functions,producesan excessof1.(d)Arandomlychosenbalancedfunction,fourelementsareover 0.5andfourunder,theexcessis1.25. 74

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Figure2-13.Thetotalexcessemittedfromeachofthe72functions.Thetwosquares indicatethetwoconstantfunctions,referringbacktoFigures 2-12 (a,b).As youcanseethebalancedfunctions(circles)separateinvegroups.The points(c)and(d)refertotherespectivegraphsofFigure 2-12 75

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CHAPTER3 SEARCHINGANUNSORTEDDATABASE Inthischapterwepresentachaoticcomputationalgorithmforsearchinganunsorted databaseforamatchbetweenaqueriedforitemandthecontentsofthedatabase[ 28 ]. ThedynamicalsystemweuseforthedemonstrationofthealgorithmistheTentmap, giveninEquation 1{3 Ingeneralmostcommonlyuseddevicesforstoringandprocessinginformationare basedonthebinaryencodingofinformation,i.e.uponbits.Largerchunksofinformation areencodedbycombiningconsecutivebitsintobytesandwords.Hereweshowadierent approachforinformationencodingandstorage,basedonthewidevarietyofpatternsthat canbeextractedfromnonlineardynamicalsystems. Wespecicallydemonstratetheuseofarraysofnonlineardynamicalsystems (orelements)tostablyencodeandstoreinformation(suchaspatternsandstrings). Furthermorewedemonstratehowthisstoragemethodenablestheecientandrapid searchforspecieditemsofinformationinthedatastore.Itisthenonlineardynamics ofeacharrayelementthatprovidesexiblecapacitystorage,aswellasthemeansto preprocessdataforexactandinexactpatternmatching.Inparticular,wechoosechaotic systemstostoreandprocessdatathroughthenaturalevolutionoftheirdynamics.More importantlyperhapswenotethat,ourmethodinvolvesjustasingleproceduralstep,itis naturallysetupforparallelimplementationandcanberealizedwithhardwarecurrently employedforchaos-basedcomputingarchitectures. Werstshowaslightlymodiedstoringandencodingscheme,basedontheExcess overowschemeofSection 2.1.1 ,inwhichweusetheactualxedpointstateofthe system,insteadoftheexcessgenerated.Specicallywedemonstratethisschemeas appliedtotheTentmapforstoring,andassociatethestoragewithmorethanjust numbers,dierentencodings.Wethenshowtheactual\searchprocess"andhowthe resultsareadirectconsequenceofthenon-linearnatureoftheTentmap.Finallywith 76

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specicexamplesweshowimplementationsofthemethodfornotonlyexactmatches,but foralso\inexact",approximatematches,toagiventarget. 3.1EncodingandStoringInformation ConsideralistofNdatastorageelements(labeledas j =1 ; 2 ;:::;N )inanarray, whereeachelementstoresandencodesofoneof M distinctitems.Ncanbearbitrarily largeand M isdeterminedbythekindofdatabeingstored.Forinstancewhenstoring Englishtextwecanconsiderthelettersofthealphabettobeeachanaturallydistinct item,so M =26.Forthecaseofdatastoredindecimalrepresentation M =10,andfor workinbioinformatics(manipulatingthesymbolsA,T,C,andG)wehave M =4.We canalsoconsiderstringsandpatternsastheitems.ForinstanceformanipulatingEnglish textwecanusealargesetofkeywordsasthebasis,necessitatingverylargeM.Westore thislistof N elementsby N dynamicallyevolvingchaoticelements.Thestateofthe elementsatdiscretetime n isgivenby X j n [ m ],where( j =1 ; 2 ;:::;N )indexeseachelement ofourlistand( m =1 ; 2 ;:::;M )indexesaniteminour"alphabet"(namelyoneofthe M distinctitems).Toreliablystoreinformationonemustconneeachdynamicalsystemto axedpointbehaviour,i.e.astatethatisstableandconstantthroughoutthedynamical evolutionofthesystemovertime n ,basicallysothatthelistremainsunchanged.We thereforeemployathresholdcontrolmechanism,seeSection 1.2.3 ,toexiblycontrolthe dynamicalelementsontothelargesetofperiod1xedpoints. Specicallyforthetentmap,thresholdsintherange[0 ; 2 3 ]yieldxedpoints,namely X n = T ,foralltime,where T isathresholdfrom0 T 2 3 .SeeFigure 3-1 fora schematicofthetentmapunderthethresholdmechanism,whichiseectivelydescribed bya\be-headedmap".ItisclearfromFigure 3-1 thatintherange[0 ; 2 3 ]thevalueof X n liesabove X n implyingthatthesystemwithstate X n atthreshold T willbemappedtoa statehigherthan T inthesubsequentiterateandthuswillbeclippedbackto T .Another wayofgraphicallyrationalizingthisistonotethatxedpointsolutionsareobtained wherethe X n +1 = X n lineintersectsthe\beheaded"tentmap.Thevalueof X atthe 77

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intersectionyieldsthevalueofthexedpoint,andtheslopeattheintersectionnaturally givesthestabilityofthexedpoint.ItisclearfromFigure 3-1 thatintherange[0 ; 2 3 ]this intersectionisonthe\plateau",namelythexedpointsolutionisequaltothethreshold value.Furtherthesolutionofthexedpointforthismapissuperstableastheslopeis exactlyzeroonthe\plateau".Thismakesthethresholdedstateveryrobustandquite insensitivetonoise. Figure3-1.TheTentmapunderthethresholdmechanism.Showntwocasesofthreshold controlat 1 4 and 1 2 .Eectivelyeachthresholdvalueisona\plateau"yielding axedpoint.Thesymbol M indicatestheactionoftheTentmaponthe thresholdedvalue X n < X n +1 ,while H indicatestheeectofthecontrol X n +1 T ,for X valuesintherange[0 ; 2 3 ]. 78

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Returningtoourdataofagiven\alphabet"wecantakealargesetofthresholds f T [1] ; T [2] ;:::; T [ M ] g fromthexedpointrange,settingupaone-to-onecorrespondence ofthese M thresholdswiththe M distinctitemsofourdata.Thisallowseachitem m to beuniquelyencodedbyaspecicthreshold T [ m ],with( m =1 ; 2 ;:::M ).Sothenumber ofdistinctitemsthatcanbestoredinasingledynamicalelementistypicallylarge,asthe sizeofMislimitedonlybytheprecisionandresolutionofthethresholdsettingandthe noisecharacteristicsofthephysicalsystembeingemployed. Thereforegivenanunsortedlistof N data,chosenoutofthe\alphabet"of M items, asdescribedabove,wesetupanarrayof N elements(labeledas j =1 ; 2 ;:::;N ),eacha Tentmap,eachwithathreshold T j [ m ]reliablystoringandencodingtheappropriateitem ofthelist.Thatis,ifelement j holdsitem m intheunsortedlist,thethresholdvalue ofelement j issetto T j [ m ],withoutchanging,orinanywayaectinganyparameter ofthelist.Sobydenotingthethresholdofelement j by T j [ m ]wehavethefollowing: ifthestateofelement j ofthesystem, X j n [ m ],exceedsitsprescribedthreshold T j [ m ] (i.e.when X j n [ m ] > T j [ m ])thestate X j n [ m ]isresetto T j [ m ].Sincethethresholdsliein therangeyieldingxedpointsofperiod1,thisenableseachelementtoholditsstateat value X j n [ m ]= T j [ m ]foralltimes n Inourencodingforareasonthatwillbecomeapparentinthenextsection,the thresholdsarechosenfromtheinterval(0 ; 1 2 ),namelyasubsetofthexedpointwindow [0 ; 2 3 ].Forspecicillustration,withoutlossofgeneralityconsidereachitemtobe representedbyaninteger m ,intherange[1 ;M ].Deningaresolution r betweeneach integeras: r = 1 2 1 ( M +1) ; (3{1) givesusalookuptable,mappingtheencodeditemtothethreshold,specicallyrelating theintegers m intherange[1 ;M ]tothethresholds T j [ m ]intherange[ r; 1 2 )]TJ/F2 1 Tf0.99999 0 Td[(r ]by: 79

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T j [ m ]= m r: (3{2) Thereforeweobtainadirectcorrespondencebetweenthesetofintegers1to M whereeachintegercanrepresentanyitem,andasetof M thresholdvaluesofadynamical system.Evenmorewecanstore N listelementsbysettingappropriatethresholds,via Equation 3{2 ,on N dynamicalelements.Asmentionedbefore,thethresholdedstates encodingdierentitemsareveryrobusttonoisesincetheyaresuperstablexedpoints. Finallythiscorrespondence,orrepresentation,isimportantfortheprocessofencoding informationinan M -levelrepresentationand,asweshallseebelow,itisprimarily importantfortheprocessofsearchingthelistforcertainbitsofinformation,byutilizinga specicpropertyofthesystem. 3.2SearchingforInformation Oncewehaveagivenliststoredbysettingappropriatethresholdson N dynamical elements,wecanqueryfortheexistenceofaspeciciteminthelist.Hereweshow howthemannerinwhichtheinformationisencodedhelpsuspreprocessthedatasuch thattheeortrequiredinthepatternmatchingsearchesisreduced.Specicallywewill demonstratehowwecanuseoneglobaloperationalsteptomapthestateofelementswith thematchingitemtoanuniquemaximalstatethatcanbeeasilydetected.Notethat suchanoperationenablesustodetectmatchestostrings/patterns(oflengthequivalent tolog 2 M binarybits)inonestep.Itwouldtaketypicallylog 2 M stepstodothesamefor thecaseofbinaryencodeddata. Whensearchingforaspeciciteminthelist,wegloballyshiftthestateofall elementsofthelistupbytheamountthatrepresentsthequerieditem.Specicallythe state X j n [ m ]ofalltheelements( j =1 ;:::;N )israisedto X j n [ m ]+ Q [ k ],where Q [ k ]isa searchkeygivenby: Q [ k ]= 1 2 )]TJ/F8 1 Tf0.99999 0 Td[(T [ k ] ; (3{3) 80

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where k istheitembeingsearchedfor,and T [ k ]itsrespectivethreshold.Thisaddition shiftstheintervalthatthelistelementscanspan,from[ r; 1 2 )]TJ/F2 1 Tf0.99276 0 Td[(r ]to[ r + Q [ k ] ; 1 2 )]TJ/F2 1 Tf0.99276 0 Td[(r + Q [ k ]], where Q [ k ]isthegloballyappliedshift.Notethatwhatwearesearchingforisthe representationoftheitem,nottheitemitself.Forexample,wecanencodeeachletterof thealphabetbyanumber,suchthatthelowestthreshold T j [1]representstheletterA,the nexthighest T j [2]representsB,etc.WhenwesearchforA,wearereallysearchingforthe elementwithastatewiththreshold T j [1]. Theitembeingsearchedforisencodedinamanner\complimentary"totheencoding oftheitemsinthelist(muchlikeakeythattsaparticularlock);i.e. Q [ k ]+ T [ k ]addsup to 1 2 .Thisguaranteesthatonlytheelement(s)matchingtheitembeingsearchedforwill haveitsstateshiftedto 1 2 .Thevalueof 1 2 isspecialinthatitistheonlystatevaluethat onthesubsequentupdateofthesystemwillreachthevalueof1 : 0,whichisthemaximum statevaluefortheTentmap.Soonlytheelementsholdinganitemmatchingthequeried itemwillreachtheextremalvalue1 : 0onthedynamicalupdatefollowingasearchquery. Notethattheimportantfeaturehereisthenonlineardynamicsmappinguniquelythe state 1 2 to1,whileallotherstates(bothhigherandlowerthan 1 2 )getmappedtovalues lowerthan1.SeeFigure 3-2 foraschematicofthisprocess. Thesalientcharacteristicofthepoint 1 2 isthefactthatitistheuniquecriticalpoint, andsoitactsas\pivot"pointforthenonlineardynamicalfoldingthatwilloccuronthe interval[ r + Q [ k ] ; 1 2 )]TJ/F2 1 Tf1.0481 0 Td[(r + Q [ k ]]duringthenextupdate.Thisprovidesuswithasingle globalmonitoringoperationtopushthestateofalltheelementsmatchingthequeried itemtotheuniquemaximalpointinparallel.Thecrucialingredientistheuseofthe existingcriticalpointinthedynamicalmappingtoimplementselection.Chaosisnot strictlynecessaryhere.Itisevidentthatforunimodalmapshighernonlinearitiesallow largeroperationalrangesforthesearchoperationandalsoenhancetheresolutionofthe encoding.FortheTentmapspecically,itcanbeshownthattheminimalnonlinearity necessaryfortheabovesearchoperationtoworkisoperationinthechaoticregion. 81

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Figure3-2.Schematicrepresentationofthechangesinthestateofanelementfor(i)a matchingquerieditem,(ii)anitem\higher"thanthequerieditem,and(iii) anitem\lower"thanthequerieditem.The\key"valueusedis Q = 1 4 ,sothe matcheditemasavalueof0 : 25.Thebehaviourofthreevaluesisshown0 : 1, 0 : 25and0 : 3.Itisclearthattheapplicationofthe\key"doesnotseemto relativelyaectthethreevalues,asimplelineartranslation.Theapplication ofthemapthough,indicatedby M ,clearlymapsboth0 : 1and0 : 3tolower statesthanthemaximalstate,acquiredsolelyby0 : 25. 82

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Anotherspecicfeatureofthetentmapisthatitspiecewiselinearityallowstheencoding andsearchoperationtobeverysimpleindeed. Ofcoursetocompletethesearchwemustnowdetectthemaximalstatelocatedat 1.Thiscanbeaccomplishedinavarietyofways.Forexample,onecansimplyemploya leveldetectortoregisterallelementsatthemaximalstate.Thiswilldirectlygivethetotal numberofmatches,ifany.Sothetotalsearchprocessisrenderedsimplerasthestate withthematchingpatternisselectedoutandmappedtothemaximalvalue,allowing easydetection.Evenmore,byrelaxingthedetectionlevelbyaprescribed\tolerance", wecancheckfortheexistencewithinourlistofnumbersorpatternsthatareclosetothe itemorpatternbeingsearchedfor.Inthiscase\closeto"means\havingarepresentation thatisclosetotherepresentationoftheitemforwhichwearesearchingfor".Usingthe earlierexampleofEnglishlettersofthealphabetencodedusingthelowestthreshold T j [1]forA,thenexthigherthresholdforB,etc.,relaxingthedetectionthresholdasmall amountallowsustondmistypedwords,whereLorNweresubstitutedforMorwhere XorZweresubstitutedforY.However,ifwehadchosenourrepresentationsuchthat theorderingputTandUbeforeandafterY(asisthecaseonastandardQWERTY keyboard),thenourrelaxedsearchwouldndspellingsof\bot"or\bou"when\boy"was intended.Thus\nearness"isdenedbythechoiceoftherepresentationandcanbechosen advantageouslydependingontheintendeduse.Figure 3-5 givesanillustrativeexampleof detectingsuchinexactmatches. Sononlineardynamicsworksasapowerful\preprocessing"tool,reducingthe determinationofmatchingpatternstothedetectionofmaximalstates,anoperationthat canconceivablybeaccomplishedbysimpleadditionandinparallel. 3.3Encoding,StoringandSearching:AnExample ConsiderthecasewhereourdataisEnglishlanguagetext,encodedasdescribed abovebyanarrayoftentmaps.Inthiscasethedistinctitemsarethelettersofthe Englishalphabet.Asaresult M =26andweobtain r = 1 54 =0 : 0185185 ::: fromEquation 83

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3{1 ,andtheappropriatethresholdlevelforeachitemisobtainedfromEquation 3{2 Morespecically,considerasourlistthesentence\ strawberryfields ".Eachletter 1 in thissentenceisanelementofthelistwithavalueselectedfromour26possiblevaluesand canbeencodedusingtheappropriatethreshold,asinFigure 3-3 (a). Nowthelist,asencodedabove,canbesearchedforspecicitems.Figure 3-3 presents theexampleofsearchingfortheletter\ l ".Todosothesearchkeyvaluecorresponding toletter\ l "(fromEquation 3{3 Q [ l ]= 15 54 )isaddedgloballytothestateofallelements. Thenthroughtheirnaturalevolution,atthenexttimestepthestateoftheelement(s) containingtheletter\ l "ismaximized.InFigure 3-4 weperformedananalogousqueryfor theletter\ e ",whichispresenttwiceinourlist,toshowthatmultipleoccurrencesofthe sameitemcanbedetected.FinallyinFigure 3-5 wesearchforanitemthatisnotpartof ourgivenlist,theletter\ x ".AsexpectedFigure 3-5 (c)showsthatnoneoftheelements aremaximized.Byloweringthedetectionleveltothevalue1 )]TJ/F0 1 Tf1.0305 0 Td[((2 r ),wehavedetected whetheradjacentitemstothequeriedonearepresent.Specicallywehavedetectedthat theletters\ w "and\ y "arecontainedinthelist.Thisdemonstratesthatinexactmatches canalsobefoundbythisscheme. 3.4Discussion Asignicantfeatureofthepresentedsearchmethodisthatitemploysasingle simpleglobalshiftoperationanddoesnotentailaccessingeachitemseparatelyatany stage.Itachievesthisthroughtheuseofnonlinearfoldingtoselectoutthematched item,andthisnonlinearoperationistheresultofthenaturaldynamicalevolutionof theelements.Sothesearcheortisconsiderablysimpliedbecauseitusesthenative responsesofthenonlineardynamicalelements.Wecanthenthinkofthisasanatural application,atthemachinelevel,inacomputingmachineconsistingofchaoticmodules [ 25 { 27 32 69 70 76 81 { 85 ].Itisalsoequallypotentasaspecial-applications\search 1 Thespacebetweenthewordsisignored. 84

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Figure3-3.Searchingfor\ l ".(a)Thresholdlevelsencodingthesentence \ strawberryfields ",barsmarkedas ;(b)thesearchkeyvalueforletter\ l isaddedtoallelements,barsmarkedas ;(c)theelementsupdatetothe nexttimestep,barsmarkedas .Forclaritywemarkedsolidblackany elementsthatreachthedetectionlevel. chip",whichcanbeaddedontoregularcircuitryandshouldproveespeciallyusefulin machines,whicharerepeatedlyemployedforselection/searchoperations. Intermsoftheprocessortimescale,thesearchoperationrequiresonedynamical step,namelyoneunitoftheprocessor`sintrinsicupdatetime.Theprincipalpointhere isthescopeforparallelismthatexistsinourscheme.Thisisduetotheselectionprocess occurringthroughoneglobalshift,whichimpliesthatthereisnoscale-up(inprinciple) withsize N .Additionallyconventionalsearchalgorithmsworkwithorderedlists,andthe timerequiredfororderinggenericallyscaleswith N as O ( N log N ).Hereincontrast,there isnoneedforordering,andthisfurtherreducesthesearchtime. Regardinginformationstoragecapacity,notethatweemployan M -stateencoding, where M canbeverylargeinprinciple.Thisoersmuchgaininencodingcapacity.As 85

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Figure3-4.Searchingfor\ e ".(a)Thresholdlevelsencodingthesentence \ strawberryfields ",barsmarkedas ;(b)thesearchkeyvalueforletter\ e isaddedtoallelements,barsmarkedas ;(c)theelementsupdatetothe nexttimestep,barsmarkedas .Forclaritywemarkedsolidblackany elementsthatreachthedetectionlevel. intheexamplewepresentabove,thelettersofthealphabetareencodedbyoneelement each;binarycodingwouldrequiremuchmorehardwaretodothesame.Specically, considertheillustrativeexampleofencodingalistofnames,andthensearchingthelist fortheexistenceofacertainname.InthecurrentASCIIencodingtechnique,eachASCII letterisencodedintotwohexadecimalnumbersor8bits.Assumingamaximumname lengthof k letters,thisimpliesthatonehastouse8 k binarybitspername.Sotypically thesearchoperationscalesas O (8 kN ).Considerincomparisonwhatourschemeoers:if base26(\alphabetical"representation)isused,eachletterisencodedintoonedynamical system(an\alphabit").Asmentionedbefore,thesystemiscapableofthisdenseencoding asitcanbecontrolledonto26distinctxedpoints,eachcorrespondingtoaletter.Again assumingamaximumlengthof k letterspername,oneneedstouse k \alphabits"per 86

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Figure3-5.Searchingfor\ x ".(a)Thresholdlevelsencodingthesentence \ strawberryfields ",barsmarkedas ;(b)thesearchkeyvalueforletter\ x isaddedtoallelements,barsmarkedas ;(c)theelementsupdatetothe nexttimestep,barsmarkedas .Itisclearthatnoelementsreachthe detectionlevelat1.0;(d)Byloweringthedetectionlevelwecandetect whetheritems\adjacent"to\ x "arepresent.Forclaritywemarkedsolidblack anyelementsthatreachthedetectionlevel(\ w "and\ y "). name.Sothesearcheortscalesas kN .Namely,thestorageis8timesmoreecient andthesearchcanbedoneroughly8timesfasteraswell!Ingeneralifbase S encoding isemployed,forexamplewhere S isthesetofallpossiblenames(size( S ) N ),then eachnameisencodedintoonedynamicalsystemwith S xedpoints(a\superbit").So oneneedstousejust1\superbit"pername,implyingthatthesearcheortscalessimply as N ,i.e.8 k timesfasterthanthebinaryencodedcase.Evenmore,inpracticethenal stepofdetectingthemaximalvaluescanconceivablybeperformedinparallel.Thiswould reducethesearcheorttotwotimesteps(onetomapthematchingitemtothemaximal valueandanothersteptodetectthemaximalvaluesimultaneously).Inthatcasethe searcheortwouldbe8 kN timesfasterthanthebinarybenchmark. 87

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Alternateideastoimplementtheincreasinglyimportantproblemofsearchhave includedtheuseofquantumcomputers[ 80 ].However,ournonlineardynamicalscheme hasthedistinctadvantagethattheenablingtechnologyforpracticalimplementation neednotbeverydierentfromconventionalsilicondevices.Namely,thephysicaldesign ofadynamicalsearchchipshouldberealizablethroughconventionalCMOScircuitry. Implementedatthemachinelevel,thisschemecanperformunsortedsearcheseciently. CMOScircuitrealizationsofchaoticsystems,likethetentmap,alreadyoperatebeyond theregionof1MHz[ 86 87 ].Thusacompletesearchforanitemcomprisingofsearchkey addition,update,thresholddetection,andlistrestorationcanbeperformedat250kHz, regardlessofthelengthofthelist.Evenmorethough,nonlinearsystemsareabundantin nature,andsoembodimentsofthisconceptcanbeconceivedinmanydierentphysical systemsrangingfromuidstoelectronicstooptics.Potentiallygoodcandidatesfor physicalrealizationoftheschemeincludenonlinearelectroniccircuitsandopticaldevices [ 88 ].Alsosystemssuchassingleelectrontunnelingjunctions[ 89 ],whicharenaturally piecewiselinearmaps,canconceivablybeemployedtomakesuchsearchdevices. Insummarywehavepresentedamethodtoecientlyandexiblystoreinformation usingnonlineardynamicalelements.Wedemonstratehowasingleelementcanstore M distinctitems,where M canbelargeandcanvarytobestsuitthenatureofthedata beingstoredandtheapplicationathand.Namely,wehaveinformationstorageelements ofexiblecapacity,capableofnaturallystoringdataindierentbasesorindierent alphabetsorwithmultilevellogic.Thiscutsdownspacerequirementsbylog 2 M in relationtoelementsstoringviabinarybits.Furtherwehaveshownhowthismethodof storinginformationcanbenaturallyexploitedforsearchingofinformation.Inparticular, wedemonstratedamethodtodeterminetheexistenceofaniteminanunsortedlist.The methodinvolvesasingleglobalshiftoperationappliedsimultaneouslytoalltheelements comprisingthelist,suchthatthenextdynamicalstep\pushes"theelement(s)storingthe matchingitem(andonlythose)toaunique,maximalstate.Thisextremalstatecanthen 88

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bedetectedbyasimpleleveldetector,directlygivingthenumberofmatches.Evenmore themaximastatecanbetreatedasamaximalrange,inwhichcaseapproximatematches areidentiedaswell. 89

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CHAPTER4 ASIMPLEELECTRONICIMPLEMENTATIONOFCHAOTICCOMPUTATION Thischapterisashortexpositionoftheresultsofourpublication[ 34 ]concerning aniteratedmapwithaverysimple(i.e.minimal)electronicimplementation.Werst proposeandcharacterizethemapandthenprovidethecircuittoimplementthemap.We proceedtodeterminecontrolthresholdsforexiblyrepresentingthevefundamentallogic gatesanddemonstratehowthismap(andcircuit)canbeusedtoimplementthesearch algorithmintroducedinChapter 3 4.1AnIteratedNonlinearMap Webeginbyconsideringaniteratedmapgovernedbythefollowingequation: x n +1 = x n 1+ x n ; (4{1) where and aresystemparameters.Figure 4-1 showsthebifurcationdiagramsfor dierentvaluesof and .Itisevidentthatthismapyieldsdynamicsrangingfromxed pointsthroughchaos.Itisalsoclearthatthemapfollowstheperiod-doublingrouteto chaoswithrespectto ,anditdoessoaswellwithrespectto .Inthefollowingsections wewillconsiderthemapinthechaoticregime,with =2and =10,namelythechaotic mapgivenby: x n +1 = 2 x n 1+ x 10 n ; (4{2) Thisoperatingpointisindicatedbythedottedlineinthebottomrightpanelof Figure 4-1 andthegraphicalformofthismapispresentedinFigure 4-2 4.2ThresholdControlChaosintoDierentPeriods UsingthemapgivenbyEquation 4{2 ,wewishtoconstructasystemthatcan represent M distinctstates,where M canbelarge.Thesizeof M willbelimitedonlyby ourabilitytodistinguishonestatefromthenextinthepresenceofnoise.Todothiswe 90

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Figure4-1.BifurcationdiagramoftheiteratedmapinEquation 4{1 forvariousvaluesof and .Thedottedlineinthebottomrightpanel,indicatesthechosen operatingpointasprescribedbyEquation 4{2 .Here x 1 isthevaluetakenby themapafterinitialtransientshavediedout. usethesimpleandeasilyimplementablethresholdcontrolmechanismdescribedinSection 1.2.3 .Specicallyweplaceundercontrolthestatevariable x as: x n +1 = 8 > > < > > : 2 x n 1+ x 10 n ,for x n +1 x x ,for x
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Figure4-2.Graphicalformofthemaptobeimplementedbyanelectroniccircuit.The parametersforthisformaresetat =2and =10. periodicitiestochaos.Asindicatedinthegure,thesystemiscontrolledtoxedpoints forthresholdsfor x < 1.Whenthethresholdisaboveunity,manydierentperiodic(as wellaschaotic)orbitsbecomeavailable. 4.3ElectronicAnalogCircuit:ExperimentalResults TherealizationofthediscretemapofEquation 4{2 incircuitryisdepictedin Figure 4-4 .Inthecircuit V in and V o denoteinputandoutputvoltagesandintermsofthe equation x n and x n +1 ,respectively.Asimplenonlineardeviceisconstructedbycoupling twocomplementary(n-channelandp-channel)(Q1,Q2)junctioneld-eecttransistors (JFETs)[ 90 ]mimickingthenonlinearcharacteristiccurve f ( x )= 2 x 1+ x 10 .Thevoltage acrossresistorR1isampliedbyafactorof5usingtheoperationalamplierU1inorder toscaletheoutputvoltagebackintotherangeoftheinputvoltage,anecessarycondition foracircuitbasedonamap.Theresultingvoltagecharacteristicsofthenonlineardevice aredepictedinFigure 4-5 ,comparewiththemathematicalformofthemapinFigure 4-2 92

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Figure4-3.Eectofthresholdvalue x onthedynamicsofthesystemgivenbyEquation 4{3 InorderthoughtorealizethemapofEquation 4{3 ,werequiretwomoresampleand holdcircuits,inadditiontoathresholdcontrollercircuit,seeFigure 4-6 .Therstsample andhold(S/H)circuitholdstheinputsignal( x n )inresponsetoaclocksignalCK1.The outputfromthissampleandholdcircuitisfedasinputtothenonlineardeviceforthe subsequentmapping,thatisEquation 4{2 .Asecondsampleandhold(S/H)circuittakes theoutputfromthenonlineardeviceinresponsetoaclocksignalCK2.Inlieuofcontrol, theoutputfromthe2 nd sample-and-holdcircuit( x n +1 )closestheloopastheinputto1 st sample-and-holdcircuit,throughthethresholdcontrolcircuit.Themainpurposeofthe twosample-and-holdcircuitsistointroducediscretenessintothesystemandadditionally 93

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Figure4-4.CircuitdiagramofthenonlineardeviceofEquation 4{3 .(Left)Intrinsic (resistorless),complementarydevicemadeoftwo(n-typeandp-type)JFETs. Q1:2N5457,Q2:2N5460.(Right)Ampliercircuitrytoscaletheoutput voltagebackintotherangeoftheinputvoltage.R1:535,U1:AD712 op-amp,R2:100kandR3:450k.Here V in = x n and V o = x n +1 tosettheiterationspeed.Toimplementthecontrolfornonlineardynamicalcomputing, theoutputfromthe2 nd sampleandholdcircuitisinputtothethresholdcontroller,as thatdescribedbyEquation 4{3 .Theoutputfromthisthresholdcontrollerthenbecomes theinputtothe1 st sample-and-holdcircuit. InFigure 4-6 ,thesampleandholdcircuitsarerealizedwithNationalSemiconductors sampleandholdICLF398,triggeredbydelayedtimingclockpulsesCK1andCK2[ 70 ]. Hereaclockrateofeither10kHzor20kHzmaybeused.Thethresholdcontrollercircuit isshowninFigure 4-7 isrealizedwithanAD712operationalamplier,a1N4148diode,a 1kseriesresistorandthethresholdcontrolvoltage, x (= V con ). TheFigure 4-8 (a)displaystheuncontrolledchaoticwaveformandtheFigures 4-8 (b-d) showrepresentativeresultsofthechaoticsystemunderdierentthresholdvalues x (= V con ).Itisclearthatadjustingthethresholdyieldscyclesofvaryingperiodicities. Also,notethatsimplysettingthethresholdbeyondtheboundsoftheattractor(5V) 94

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Figure4-5.Voltageresponsecharacteristicsofthenonlineardevice,basedonEquation 4{2 andcircuitofFigure 4-4 givesbacktheoriginaldynamics,andsothecontrolleriseasilyswitchedonando.A detailedcomparisonshowscompleteagreementbetweenexperimentalobservationsand analyticalresults.Forinstance,thethresholdthatneedstobesetinordertoobtaina certainperiodicityandthetrajectoryofthecontrolledorbitcanbeworkedoutexactly throughsymbolicdynamicstechniques.Further,thecontroltransienceisveryshorthere (typicallyoftheorderof10 )]TJ/F5 1 Tf0.8264 0 Td[(3 timesthecontrolledcyclelength)andtheperturbation involvedinthresholdcontrolisusuallysmall.Thismethodisthenespeciallyusefulin thesituationwhereonewishestodesigncontrollablecomponentsthatcanswitchexibly betweendierentbehaviours.Calibratingthesystemscharacteristicsattheoutsetwith 95

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Figure4-6.Schematicdiagramforimplementingthethresholdcontrollednonlinearmap. CK1andCK2areclocktimingsignals,whilethemodulesdesignatedS/Hare sample-and-holdcircuits. Figure4-7.Circuitdiagramofthethresholdcontroller. V in and V o aretheinputand output,Disa1N4148diode,R=1k,andU2isanAD712op-amp. V con = x (controllerinputvoltage). 96

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respecttothresholdgivesonealook-uptabledirectlyandsimplytoextractwidely varyingtemporalpatterns. Figure4-8.PSPICEsimulationresultsoftheexperimentalcircuit.Theordinateis x n and theabscissaisthediscretetime n measuredinms.(a)Uncontrolledchaos: x =6V,(b)period5cycle: x =4V,(c)period2cycle: x =3 : 7Vand(d) period1cycle: x =3 : 5V. 4.4FundamentalLogicGateswithaChaoticCircuit Hereweexplicitlyshowhow,byusingthethresholdcontrolledmapofEquation 4{3 weobtainstheclearlydenedlogicgateoperationsNOR,NAND,AND,OR,and XOR.Thestateofthesystemisrepresentedbythestatevalueof x .Theinitial stateofthesystemisrepresentedas x 0 .Inourmethodallvebasicgateoperations involvethefollowingsteps:specicationof x 0 basedontheoperationandtheinputs throughthresholdcontrol,nonlinearupdate(evolutionofthecircuitdynamics),and outputinterpretationthroughthreshold\monitoring"inthespiritofExcessOverow Propagation,fromSection 1.2.3 .Specically: 1.InputsandProgramming; x 0 = x prog + x I 1 + x I 2 .Here x prog isaprogramming shiftthatxestheinitialstate x 0 ofthesystem,basedonthegatetobeoperated. Lettinganitevoltage denotealogical1,weset x I i = foraninputoflogical1 and x I i =0foraninputoflogical0. 97

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2.Nonlinearupdate;i.e., x 0 f ( x 0 ),where f ( x )isthenonlinearfunction,givenby Equation 4{2 3.ThresholdingtoobtaintheoutputZdenedas: Z= ( 0,for f ( x ) x f ( x ) )]TJ/F2 1 Tf0.99999 0 Td[(x ,for x
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Table4-2.Necessaryandsucientconditionstobesatisedbyachaoticelementinorder toimplementthelogicaloperationsNOR,NAND,AND,ORandXOR.Where f ( x )isgivenbyEquation 4{2 and1isthemonitoringthreshold( x )of interpretationofthelogic output. I 1 I 2 NORNAND AND 0 01
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elementtorecongureintodierentlogicgatesthroughathresholdbasedmorphing mechanism.Contrastthistoaconventionaleldprogrammablegatearrayelement,where recongurationisachievedthroughswitchingbetweenmultiplesinglepurposegates. 4.5EncodingandSearchingaDatabaseUsingChaoticElements InthespiritofChapter 3 weapplythatmethodtothemap,andcircuit,givenby Equation 4{3 .Specicallyweshowhowthismapcanbeutilizedtostablyencodeand storevariousitemsofinformation(suchaspatternsandstrings)tocreatea\database". Furtherwedemonstratehowthisstoragemethodallowstoecientlydeterminethe numberofmatches(ifany)tosomespecieditem[ 28 ].Consideranarrayofelementseach ofwhichevolvesaccordingtoEquation 4{3 .Thenonlineardynamicsofthearrayelements willbeutilizedforexiblecapacitystorage,aswellasforpre-processingdataforexact (andinexact)patternmatchingtasks. Encodinginformation .Weconsidera\database"oflength N andeachmember ofthedatabaseisencodedinanelementobeyingEquation 4{3 ,weindextheseelements with j = f 1 ; 2 ; 3 ;:::N g ,sothestateofthewholearray,atatime n ,canberepresented as X j n .Atthesametimethedatabaseismadeupofitemsfroman\alphabet"oftotal numberofuniqueitems M ,indexedwith m = f 1 ; 2 ; 3 ;:::M g .Wecorrelateeachitem m withathreshold, x ,forEquation 4{3 ,suchthattheelementisconnedonaxedpoint ofperiod1,wedene T j [ m ]asthethresholdforthe j th elementencodingthe m th item. Forthismap,thresholdsrangingfrom0to1yieldxedpoints,asdepictedinFigure 4{3 Namely X j n = T j [ m ],foralltime n ,whenthethresholdischosenas0
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Inourencoding,thethresholdsarechosenfromtheinterval(0 ; 1 2 ),namelyasub-set ofthexed-pointwindow(0,1) 1 .Withoutlossofgenerality,considereachitemtobe representedbyaninteger z fromtherange[1 ;M ].Deningaresolution r betweeneach thresholdas: r = 1 2 1 M ; (4{4) givesalookupmapfromtheencodedintegertothethreshold,relatingtheintegers z in theset[1 ;M ]tothresholds T j [ m ]intherange[ r; 1 2 ],by: T j [ m ]= z r: (4{5) Thereforeweobtainadirectcorrespondencebetweenasetofintegersrangingfrom 1to M ,whereeachintegerrepresentsanitem,andasetof M thresholdvalues.Sowe canstore N databaseelementsbysettingappropriatethresholds(viaEquation 4{5 )on N dynamicalelements.ClearlyfromEquation 4{5 ,ifthethresholdsettinghasbetter resolution(smaller r ),thenalargerrangeofvaluescanbeencoded.Notehoweverthat precisionisnotarestrictiveissuehere,asdierentdatarepresentationscanalwaysbe choseninordertosuitagivenprecisionofthethresholdmechanism. ProcessingInformation .Oncewehaveagivendatabasestoredbysetting appropriatethresholdson N dynamicalelements,wecanqueryfortheexistenceofa speciciteminthedatabaseinoneglobaloperationalstep.Thisisachievedbyglobally shiftingthestatevariableofallelementsofthedatabaseupbyanamountthatrepresents theitembeingsearchedfor. 1 Actuallywecanuseasmuchastheinterval(0 ; 0 : 8027),since0.8027isthepre-image ofthemaximum. 101

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Notingthatthemaximalstatevariablevalueforthissystemis1.4449,oneraisesthe state X j n ofeachelement j to X j n + Q [ k ],where Q [ k ]isasearchkeygivenby: Q [ k ]=0 : 8027 )]TJ/F2 1 Tf0.99999 0 Td[(T [ k ] ; (4{6) where k istheindexoftheinteger(item)beingqueriedfor, k 2 z .0.8027istheunique valueofthissystemthatevolvestothemaximalvalue1.4449onaniterationofthe system.Sothevalueofthesearchkeyissimply0.8027(thepre-imageofthemaximal statevariablevalue)minusthethresholdvaluecorrespondingtotheitembeingsearched for,given k 2 z wehave T [ k ]= k r Theadditionofthe\searchkey", Q [ k ]shiftstheintervalthatthedatabaseelements canspan,from[ r; 1 2 ]to[ r + Q [ k ] ; 1 2 + Q [ k ]].Since Q [ k ]+ T [ k ]addsupto0.8027,it isguaranteedthatonlytheelement(s)matchingtheitembeingqueriedforwillhave its(their)stateshiftedto0.8027,whichistheonlystatewhichafterthesubsequent iterationwillmaximizetothevalueof1.4449 2 .Sothetotalsearchprocessisrendered simpleasthestatewiththematchingpatternisselectedoutandmappedtothemaximal value,allowingeasydetection.Further,byrelaxingthedetectionlevelbyaprescribed \tolerance",wecancheckfortheexistencewithinourdatabaseofnumbersorpatterns thatare\closeto" 3 RepresentativeExample .ConsiderthecasewhereourdataisEnglishlanguage text,encodedasdescribedaboveonaletterbyletterbasisbyanarrayofmaps,following Equation 4{5 .InthiscasethedistinctitemsarethelettersoftheEnglishalphabetand wehaveM=26.Weobtain r = 1 52 0 : 0192fromEquation 4{4 andtheappropriate thresholdlevelforeachitemisobtainedviaEquation 4{5 .Moreconcretely,consideras 2 Noteallotherstates(bothhigherandlowerthan0.8027)getmappedtovalueslower than1.4449. 3 Where\closeto"isdenedbythedesignerofthedatabase. 102

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ourdatabasethephrase\ thequickbrownfox ";eachletterinthisphraseisanelementof thedatabaseandcanbeencodedusingtheappropriatethreshold,asinFigure 4-9 (a). Figure4-9.Searchingfor\ b ".(a)Thresholdlevelsencodingthephrase \ thequickbrownfox ",barsmarkedas ;(b)thesearchkeyvalueforthe letter\ b "isaddedtoallelements,barsmarkedas ;(c)theelementsupdate tothenexttimestep,barsmarkedas .Forclaritywemarkblackthe elementsthatreachedthedetectionlevel. Nowwequerythedatabaseregardingtheexistenceofspecicitems.Figure 4-9 presentstheexampleofqueryingfortheletter\ b ".Todosothesearchkeyvalue correspondingtoletter\ b "( 2 52 )isaddedgloballytothestatesofallelements,Figure 4-9 (b).Thenthroughtheirnaturalevolution,uponthenexttimestep,thestate(s)ofthe element(s)containingtheletter\ b "is(are)maximized,Figure 4-9 (c).InFigure 4-10 we performananalogousqueryfortheletter\ o ",whichhappenstobepresenttwiceinour databasetoshowthatmultipleoccurrencesofthesameitemcanbedetected.Finallyin Figure 4-11 wequeryforanitemthatisnotpartofourgivendatabase,theletter\ d ".As expectedFigure 4-9 (c)showsthatnoneoftheelementsaremaximized.Byloweringthe 103

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Figure4-10.Searchingfor\ o ".(a)Thresholdlevelsencodingthephrase \ thequickbrownfox ",barsmarkedas ;(b)thesearchkeyvalueforthe letter\ o "isaddedtoallelements,barsmarkedas ;(c)theelements updatetothenexttimestep,barsmarkedas .Forclaritywemarkblack anyelementsthatreachedthedetectionlevel. detectionleveltothevalue1 : 4449 )]TJ/F2 1 Tf0.97056 0 Td[(f (0 : 8027 )]TJ/F2 1 Tf0.97056 0 Td[(r )=1 : 4411,just\onestep"downfromthe maximal,wedetectwhetheritems\adjacent"tothedesiredonearepresent.Specically wedetectthattheletters\ c "and\ e "arecontainedinourdatabase.Thisdemonstrates thatinexactmatchescanalsobefound,justaseasily. 4.6Conclusion Insummary,weintroducedasimplemaphavingrichnonlineardynamics,and asimpleelectroniccircuitrealization.Thenwedemonstratedthedirectandexible implementationofthevebasiclogicgatesusingthissimplenonlinearmap(circuit). Further,weshowedhowthedynamicsofthismapcanbeutilizedtoprovideanecient databasesearchmethod.Wehaveexperimentallyimplementedtheelectroniccircuitanalog 104

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Figure4-11.Searchingfor\ d ".(a)Thresholdlevelsencodingthephrase \ thequickbrownfox ",barsmarkedas ;(b)thesearchkeyvalueforthe letter\ d "isaddedtoallelements,barsmarkedas ;(c)theelements updatetothenexttimestep,barsmarkedas .Itisclearthatnoelements reachthedetectionlevelat1.4449;(d)Byloweringthedetectionlevelwecan detectwhetheritems\adjacent"to\ d "arepresent(\ c "and\ e "). ofthisnonlinearmapandhavedemonstratedtheecacyofthethresholdcontrollerin yieldingdierentcontrolledresponsesfromthismapcircuit. 105

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CHAPTER5 LOGICOPERATIONSFROMEVOLUTIONOFDYNAMICALSYSTEMS Inthischapterweproposethedirectandexibleimplementationoflogicoperations usingthedynamicalevolutionofanonlinearsystem[ 33 ].Theconceptinvolvesthe observationofthestateofthesystematdierenttimeinstancestoobtaindierentlogic outputs.WeexplicitlyimplementthebasicNAND,AND,NOR,ORandXORlogicgates, aswellasmultiple-inputXORandXNORlogicgates.Furtherwedemonstratehowthe singledynamicalsystemcandomorecomplexoperationssuchasbitbybitadditionin justtwoiterations.Theconceptusesthenonlinearcharacteristicsofthetimedependence ofthestateofthedynamicalsystemtoextractdierentresponsesfromthesystem.The highlightofthismethodisthatasinglenonlinearsystemiscapableofyieldingatime sequenceofdierentlogicoperations.Furtherweexplicitlydemonstrate,throughthethree examples,howresultsfromthismethodcanbeobtainedbyvaryinganyofthe\dening variables"( x 0 x init n ). 5.1GenerationofaSequenceof(2-input)LogicGateOperations Weoutlineamethodforobtainingthevebasiclogicgatesusingdierentdynamical iteratesofasinglenonlinearsystem.Inparticularconsiderachaoticsystemwhosestate isrepresentedbyavalue x .Thestateofthesystemevolvesaccordingtosomedynamical rule.Forinstance,theupdatesofthestateoftheelementfromtime n to n +1maybe welldescribedbyamap,i.e., x n +1 = f ( x n ),where f ( x )isanonlinearfunction.Nowthis elementreceivesinputsbeforetherstiteration(i.e.,at n =0)andoutputsa\signal" afterevolvingfora(short)speciedtimeornumberofiterations.Themethodcanbe appliedforanysequenceofthegates,forillustrativepurposeswechosethesequence NAND,AND,NOR,XORandOR(seeTable 5-1 forthetruthtable). Ingeneralthemethodinvolvesthefollowingsteps: 1.Inputdenition(fora2inputoperation): x 0 = x init + x I 1 + x I 2 ,where x init isthe initialstateofthesystem.(Comparableto x prog inpreviouschapters,butnowit isnotdeningasinglegatebutasequenceofgates,ormoregeneral,operations). 106

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Table5-1.ThetruthtableofthevebasiclogicoperationsNAND,AND,NOR,XOR, OR. I 1 I 2 NANDANDNORXOR OR 0 01010 0 0 11001 1 1 01001 1 1 10100 1 beforedatainputsareintroduced, x 0 istheactualinitialstateofthesystemthat includesthedatatobeoperatedon.Aspreviouslyweset x I i = forthelogical input I i =1,and x I i =0forthelogicalinput I i =0.Soweneedtoconsiderthe followingthreecases: (a)Both I 1 ,and I 2 are0(row1inTableI),i.e.theinitialstateofthesystemis: x 0 = x init +0+0= x init (b)Either I 1 =0and I 2 =1,or I 1 =1and I 2 =0(row2or3inTableI),i.e.the initialstateis: x 0 = x init +0+ (c)Both I 1 and I 2 are1(row4inTableI),i.e.theinitialstateis: x 0 = x init + + = x init +2 2.Chaoticevolutionoversomeprescribednumberofsteps,i.e. f n ( x 0 ) x n ,for1 x n ,where x n isa referencemonitoringvalue,attimeinstance n Sincethesystemischaotic,inordertospecifytheinitial x 0 accuratelyweemploy thethresholdcontrolmechanism,seeSection 1.2.3 ,wenotethatthismechanismcanbe invokedatanysubsequentiterationaswell.Forlogicrecovery,theupdatedorevolved valueof f ( x )iscomparedwith x n valueusingacomparatoraction,inthespiritofExcess OverowPropagation,againasinSection 1.2.3 Inordertoobtainallthedesiredinput-outputresponsesofthedierentgates,as displayedinTable 5-1 ,weneedtosatisfytheconditionsenumeratedinTable 5-2 .Note thatthesymmetryofinputstooutputsreducesthefourconditionsinthetruthTable 5-1 tothreedistinctconditions,with2 t extnd and3 rd rowofTable 5-1 leadingtothe2 nd conditionofTable 5-2 107

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Table5-2.Necessaryandsucientconditionstobesatisedbyachaoticelementinorder toimplementthelogicoperationsNAND,AND,NOR,XORandORon subsequentiterations.Here x init =0 : 325and =0 : 25,forall n .While x n =0 : 75,for n = f 1 ; 2 ; 3 ; 4 g ,thatisforNAND,AND,NOR,XORlogic operations,and x 5 =0 : 4forORlogicop eration. Logic gate NAND AND NORX OR OR Iteration (n) 1 2 3 4 5 Condition 1: x 1 = f ( x 0 ) >x 1 f ( x 1 ) x 2 f ( x 2 ) >x 3 f ( x 3 ) x 4 f ( x 4 ) x 5 Logic input (0,0) x 1 =0 : 88 x 2 =0 : 43 x 3 =0 : 98 x 4 =0 : 08 x 5 =0 : 28 x 0 =0 : 325 Condition 2: x 1 = f ( x 0 ) >x 1 f ( x 1 ) x 2 f ( x 2 ) x 3 f ( x 3 ) >x 4 f ( x 4 ) >x 5 Logic input (0,1)or (1,0) x 1 =0 : 9775 x 2 =0 : 088 x 3 =0 : 33 x 4 =0 : 872 x 5 =0 : 45 x 0 =0 : 325+ x 0 =0 : 575 Condition 3: x 1 = f ( x 0 ) x 1 f ( x 1 ) >x 2 text f ( x 2 ) x 3 f ( x 3 ) x 4 f ( x 4 ) >x 5 Logic input (1,1) x 1 =0 : 58 x 2 =0 : 98 x 3 =0 : 1 x 4 =0 : 34 x 5 =0 : 9 x 0 =0 : 325+2 x 0 =0 : 825 Sogivendynamics f ( x ),wemustndvaluesofthreshold(s) x n ,initialstate(s) x init and satisfyingtheconditionsderivedfromthespecictruthtabletobeimplemented. Using,asusual,theLogisticmap,Equation 1{1 ,weincorporatedinTable 5-2 actual valuesof x init and x n for n =1 ; 2 ; 3 ; 4 ; 5,whichsatisfytheconditionsimposedbythetruth tableconsidered.Forillustrativepurposes,thegraphicalrepresentationofveiterationsof theLogisticmapisshowninFigure 5-1 ,displayingtheresultsinTable 5-2 Insummary,theinputssetuptheinitialstate x init + x I 1 + x I 2 .Thenthesystemevolves over n iterativetimestepstoeachupdatedstate x n .Theevolvedstateiscomparedtoa monitoringthreshold x n ,atevery n .Ifthestateatiteration n ,isgreaterthanthe thresholdalogical1istheoutputandifthestateislessthanthethresholdalogical0is theoutput.Thisprocessisrepeatedforeachsubsequentiteration. Notethatintheaboveexamplewepresentresultsforwhich x init isnotvariedand x n isvaried;theemphasisisonthespecicbehaviourthatrepresentsacomputational 108

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Figure5-1.GraphicalrepresentationofveiterationsoftheLogisticmap.Threedierent initialconditionsareconsidered,eachrepresentingoneofthethreecasesof twologicinputs,(0,0)by N ,(0,1)/(1,0)by N ,(1,1)by M .Ateachiteration comparisonwithamonitoringthresholdisperformed,0.75for n = f 1 ; 2 ; 3 ; 4 g 0.4for n =5.Theresultsfromtwogatesarealsoshown,thecirclesmark n =1(theNANDgate)comparewith0.75andthecolouringcorrespondsto theappropriatecaseofdata;thesquaresmark n =5(theORgate)compare with0.4,andagainthecolouringisappropriatetothedata. 109

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taskandnotthe\actual"valuesassuch.Thecaseof isslightlymorecomplicatedsince itactuallyrepresentsaconstant\entity",neverthelessvaryingits\actual"valueisalso possible,butmorecareisneeded,henceinthisexamplewesimplyset =0 : 25. Thereforeinamoregeneralcontext,wearerelatinginputsandrequiredoutputs withspecicbehaviouralpatterns,atthesametimewecansaythereverse,wendthe behaviouralpatternthatcanrepresentaneededoperation.Hencethe\actual"state valuesarenotofgreatimportance,andwecangenerate\templates"likeFigure 5-2 Togeneratethisspecictemplate,weset x n =0 : 65andvaried x 0 ,incontrasttothe exampleabove;onceagain waskeptconstantat0.25.Basicallythebehaviourofeach combinationofvariablesistheninterpretedasoneoftheeightpossiblesymmetricbinary operations.Itisclearfromthisgurethatwearenotconnedto n< 5,sothe\length" ofthesequenceofoperationscanbeextended,andalsotheactualorderofthesequence ofoperationscanbechanged.Theoreticallyalloperationsandallsequencesofoperations exist,undersomecombinationof\actual"valuesofvariables. Weshouldnotethat,asisclearfromFigure 5-2 ,therangeofeachoperationis decreasinginsizewithincreasingiterations,andsincethedynamicsarechaoticat somepointwewilllosedenition,aswementionedabovethoughthethresholdcontrol mechanismcanbeinvokedto\re-initialize"thesystem. 5.2TheFullAdderand3-InputXORandNXOR Thissectionisadirectextensionoftheprevioussection,henceitiscomprisedof simplytwodemonstrations.Weextendtheabovemethodforsequentiallogicoperations intwoways,rsttomorethanjusttwodatainputs,andsecondtomorethanjustlogic gateoperations.Specicallyweshowtheimplementationofthebinaryfulladderandthe implementationsof3-inputXORandNXORgates. Fortheseimplementationsweemploytheusualthreesteps,butmodiedfor3inputs asfollows: 110

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Figure5-2.Patternsofbinarytwoinputsymmetricoperations.WeusetheLogisticmap with =0 : 25, x n =0 : 65 ; 8 n andvary x init 1.Inputdenition(fora3inputoperation): x 0 = x init + x I 1 + x I 2 + x I 3 ,asusualwith theadditionofone\extra"input.Inthecontextofthefulladder, I 1 corresponds totheinputbinarynumberA, I 2 correspondstotheinputbinarynumberB,and I 3 correspondstothecarryinput C in (thecarryfromthe\previous"positionaldigit addition),asinTable 5-3 .Soweneedtoconsiderthefollowingfourcases: (a)Ifallinputsare0(1 st rowinTable 5-3 ),i.e.theinitialstateofthesystemis: x 0 = x init +0+0+0= x 0 (b)Ifanyoneoftheinputequals1(2 nd ,3 rd and5 th rowinTable 5-3 ),i.e.the initialstateis: x 0 = x init +0+0+ = x init +0+ +0= x init + +0+0= x init + (c)Ifanytwoinputsequalto1(4 th ,6 th and7 th rowinTable 5-3 ),i.e.theinitial stateis: x 0 = x init +0+ + = x init + +0+ = x init + + +0= x init +2 111

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(d)Ifallinputsequalto1(8 th rowinTable 5-3 ),i.e.theinitialstateis: x 0 = x init + + + = x init +3 2.Chaoticevolution(fortwotimesteps),oftheinitialstategivenfromthedata f 2 ( x 0 ) x 2 .Weconnethetimestepstoonlytwotomakecleartheexampleofthe fulladder.Ingeneralofcourse,furthertimestepscanbealsoconsideredproviding resultsforother3inputoperations,asinFigure 5-2 3.Theevolvedstate f n ( x 0 )yieldsthelogicoutputasfollows: LogicOutput=0,if f n ( x 0 ) x n LogicOutput=1,if f n ( x 0 ) >x n where x n isamonitoringthreshold,with n =1 ; 2inthefulladderexample. Table5-3.Thetruthtableoffulladder,andnecessaryconditionstobesatised.Using theLogisticmap,statevalues x 1 (iteration n =1)and x 2 (iteration n =2)are usedtoobtain C out and S respectively.Here x 1 =0 : 8, x 2 =0 : 4, x init =0 : 0and =0 : 23. Inputbit ofInputbit ofInputbit of OutputConditionfor Output Number (A)Number (B)Carry( C in ) C out SC out ( x 1 ) S ( x 2 ) 0 0 0 0 0 f ( x 0 ) x 1 f ( x 1 ) x 2 0 0 1 0 1 f ( x 0 + ) x 1 f ( x 1 ) >x 2 0 1 0 0 1 f ( x 0 + ) x 1 f ( x 1 ) >x 2 0 1 1 1 0 f ( x 0 +2 ) >x 1 f ( x 1 ) x 2 1 0 0 0 1 f ( x 0 + ) x 1 f ( x 1 ) >x 2 1 0 1 1 0 f ( x 0 +2 ) >x 1 f ( x 1 ) x 2 1 1 0 1 0 f ( x 0 +2 ) >x 1 f ( x 1 ) x 2 1 1 1 1 1 f ( x 0 +3 ) >x 1 f ( x 1 ) >x 2 TheFullAdder .Wenowdemonstratehowonecanobtaintheubiquitousbit bybitarithmeticaddition,involvingthreelogicinputsandtwooutputs,inconsecutive iterations,withasingleone-dimensionalchaoticelement.Ingeneralthesimple1-bit binaryarithmeticadditionrequiresafulladderlogicwhichaddsthreeindividualbits together(twobitsbeingthedigitinputsandthethirdbitassumedtobecarryfrom theadditionofthenextleast-signicantbitadditionoperation,knownas\ C in ").A typicalfull-adderrequirestwohalf-addercircuitsandanextraXORgate.Intotal,the implementationofafull-adderrequiresvedierentgates(threeXORgatesandtwo ANDgates).Howeverinthepresentdirectimplementationbyutilizingthedynamical evolutionofasinglemap,theLogisticmap,weneedonlytwoiterationstoimplementa 112

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fulladder.Thetruthtableandnecessaryconditionstobesatisedfortheparallelized fulladderoperationaregiveninTable 5-3 .TheCarrybitoutput C out andtheSumbit output S arerespectivelyrecoveredfromtherstandseconditerationsofaLogisticmap, with =0 : 23, x init =0 : 0 x 1 =0 : 8and x 2 =0 : 4.Noteinthisexamplewehavekept x init constantandvariedthe x n Sobasicallyeachsystemtakesthethreeinputs,thevalueofthetwobitstobeadded andthecarryfromthepreviousbitaddition,andproducesthecarryforthenextaddition ontheveryrstupdate.Thisnewcarrycanofcoursebeimmediatelysuppliedtothenext systemreadytoperformtheadditionofthenextbit,whilethesum( S )is\calculated"on thesecondupdate. 3-InputXORandNXOR .Threeormoreinputlogicgatesareadvantageous becausetheyrequirelesscomplexityinactualexperimentalcircuitrealizationthanthatof couplingconventional2-inputlogicgates[ 91 92 ].Specicallythetruthtablefor3-input XORandXNORlogicgateoperations,andthenecessaryconditionstobesatisedby amaptoperformtheseoperations,areshowninTable 5-4 .Inthisrepresentativecase, weconsideradierentscenario,forbothoperationsweusethe2 nd iterationofthemap, sobothoperationscannotbeaccomplishedbythesamesystem.Wesetthethreshold ( x 2 )atthexedvalueof0.5andsoweaccomplishtheXORgatewith x init =0 : 0and theNXORgatewith x init =0 : 25,mostimportanlythoughwehaveanexactxedvalued 0 : 25. 5.3Conclusion Ourpreviousresultshaveshownthatasinglenonlineardynamicalsystem,with propertuningofparametersandcontrolinputs,canexiblybecomeanylogicgate.We didconneourselvesthough,togatesofonlytwoinputsandtopatternscreatedbya singleiterationofasystem. Themainresultofthischapteristhatthevarioustemporalpatternsembeddedin thedynamicalevolutionofnonlinearsystemsarecapableofperformingsequencesof 113

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Table5-4.Thetruthtableofthe3-inputXORandNXORlogicoperations,necessary andsucientconditionstobesatisedbythemap.Statevalue x 2 (iteration n =2)isusedforlogicoperationrecovery.Here x =0 : 5and 0 : 25. I 1 I 2 I 3 XOR( I 1 I 2 I 3 )XNOR( I 1 I 2 I 3 )XOR( x init =0 : 0)NXOR( x init =0 : 25) 0 0 0 0 1 x 2 x x 2 )]TJ/F2 1 Tf0.99998 0 Td[(x 0 0 1 1 0 x 2 )]TJ/F2 1 Tf0.99999 0 Td[(x x 2 x 0 1 0 1 0 x 2 )]TJ/F2 1 Tf0.99999 0 Td[(x x 2 x 1 0 0 1 0 x 2 )]TJ/F2 1 Tf0.99999 0 Td[(x x 2 x 0 1 1 0 1 x 2 x x 2 )]TJ/F2 1 Tf0.99998 0 Td[(x 1 0 1 0 1 x 2 x x 2 )]TJ/F2 1 Tf0.99998 0 Td[(x 1 1 0 0 1 x 2 x x 2 )]TJ/F2 1 Tf0.99998 0 Td[(x 1 1 1 1 0 x 2 )]TJ/F2 1 Tf0.99999 0 Td[(x x 2 x logicoperationsintime(oriterates),andminimizethecontrolthatisneeded,weonly invokecontrolmechanismoninitialization,fromthereonwejustmonitorthestate.The implementationofasequenceoflogicfunctionsintime,asdescribedabove,isanother mechanismthroughwhichcomputerarchitecturesbaseduponthechaoscomputing approachcanbeoptimizedforbetterperformance.Evenfurtherinthischapterwe introducedmechanismsformulti-inputlogic,andshowedhowinthecontextofchaotic computationbehavioursaremoreimportantthanexactandspecicvaluesofvariables, providingevenmoreexibilityinaphysicalrealization. Withthesefundamentalingredientsinhanditbecomesclearthatexploitingnotjust thepatternformationofnonlineardynamicalsystems,buttheformationofsequencesof suchpatterns,producednaturallybysuchsystems,mayproveakeyingredienttowards makingnonlineardynamicalcomputationalarchitecturesarealalternativetoconventional staticlogiccomputerarchitectures. 114

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CHAPTER6 MANIPULATINGTIMEFORCOMPUTATION WehaveshowninChapter 5 howtimecanbeinvolvedinthecomputationalprocess, specicallywehaveshownhowspecic\timeinstances"canbeusedforprogramming asystemtoperformaspeciccomputationaloperation.Inthepresentchapter,insome senseweextendtheutilityoftime;weshowhowthedimensionoftimecanbeusedfor acompletedenitionofacomputationaloperation,thatisweusetimeinstancesnot onlyforprogrammingasystem,butalsoforintroducingtothesystemthe\data"to bemanipulated.Toachievethiswepredenethebehaviourofthestateofthesystem withina\clockcycle"andworkwiththefactthatthisbehaviourisnotconstantwithin thegivencycleandsoatdierenttimeinstancesdierentresultscanbeobtained.In asimpliedmathematicalcontextwearejustinterchanging x with t inthealready presentedalgorithms. 6.1Introduction Werstpresent,inageneralcontext,thetimebasedalgorithmforperforming thevefundamental2-inputlogicgatesandthenthealgorithmforperformingthe searchofanunsorteddatabase.Furtherwesupportthemethodwithspecicillustrative examplesusingneuralsystems[ 93 94 ],andelectroniccircuits.Weshouldnotethatin bothexamplesthegeneralmethodisadaptedtobestsuitthecontextofeachtheactual implementation. 6.1.1FlexibleLogicGates Foranycomputationalsystemtobeabletoperformexible(2-input)logicthree inputsneedtobegiventothesystem.Therstinputisthe\ProgrammingInstruction" thatdeneswhichlogicoperationwillbeperformed.Theothertwoinputsarethetwo logicvariables,\Input1"( I 1 )and\Input2"( I 2 ),seeFigure 6-1 .Inthepresentmethod weutilizeatimeinstance, t total ,asasingleadjustablevariablethatcanbeusedfor therepresentationofallthreeinputs.Wedeneonetimeinstancetorepresentthe 115

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Figure6-1.Schematicrepresentationofaexible2-inputlogicgate.ForanOutputthree parametersneedtobedened,thetwodatastreams,Input1andInput2,and oneProgrammingInstruction,whichistomanipulatethedata. programming, t prog ,andtwomoretimeinstancestorepresentthetwoinputs, t I 1 t I 2 respectively,withtherelationshipforatotaltimedenedas: t total = t prog + t I 1 + t I 2 .We representalogical1byanitetimelength, t in (relateto inthecaseofalgorithmsbased onstatemanipulation),andalogical0byatimelengthofzerolength( t in =0). Itisimportanttonotethat: Sincealogical0attheinputsissettobeofzerotimelength,thesetofinputs(0,0) isthesametimeintervalasprogrammingthegate( t prog ). Thetimeintervalforrepresentinganinputoflogical1mustbethesameirrespective ofwhetherthelogical1isatInput1oratInput2,asthetwoareindistinguishableby alogicgate;thesets(1,0)and(0,1)arethusrepresentedbythesametimelength; Thetotaltimeintervalforthesetofinputs(1,1)istwicethatofthesets(1,0)and (0,1). 116

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Thewayweusethetotaltime, t total ,istodeneatimeinstancewithintheclock cycleofa\genericsignal"constructedinsuchawaysoastoencompasstheresultsof dierentlogicoperationsatdierenttimeinstances.Ofcoursebecauseofthe\nature" oftimethistimeinstanceneedstobeinreferencetoaxed\zeroth"timeinstance, whichcanbedenedbytheinitializationoftheclockcycle.Infactinthegeneralcontext demonstrationweconsiderasinglecycleinitiatedatthezerothinstanceanduse t total to denea\momentofobservation",atwhichtheresultisobtained.Thereisthoughthe conversescenarioaswell,whichweshowinafollowingillustrativeexampleinSection 6.2 wecanconsiderthe\nal"timeinstanceasxedanduse t total asa\negative"shiftin instancesfromthisnalinstance,basicallychangingthe\momentofgeneration"ofthe genericsignal. Toconstructthe\genericsignal"weconsiderthebehaviourofthevefundamental logicgates(AND,NAND,OR,NOR,XOR).Fromthetruthtablefortheselogicgates (Table 6-1 )wecancreateforeachgateagraphoftheinput-outputrelationshipsforthe threepairsetsoflogicalinputs, f (0 ; 0) g f (1 ; 0) = (0 ; 1) g f (1 ; 1) g ,Figure 6-2 (a)-(e).Since wearetoconsidertimeastheinputquantity,wecanencapsulateallvegraphsintoa singlegraphwheretheinputdimensionisidentiedwithtimeinstances,asexplainedin Figure 6-2 (f). Thecombinationofthetruthtablesintothissinglenon-linearfunctionresultsinthe formof: f ( u;t ) > 0 ; for t )]TJ/F0 1 Tf1.0032 0 Td[( t>t>t + t; else f ( u;t )=0;whichprovidestherequired behaviourforasystemtobeutilizedasaexiblelogicgate.Thisfunctionisverysimilar toanactionpotentialgeneratedbyaneuron;restingatalowvoltage( f ( u;t )=0),and forabrieflengthoftimerisingveryrapidlytoahighervoltage( f ( u;t ) > 0),andthen droppingveryrapidlybacktoitsoriginallevel.Weareconsideringthisfunctioninneural context,notonlybecauseoftheformofthefunction,butalsobecauseitisclearthat insomewayneuronsarecapableoflogicoperations[ 95 ].Notethough,thatwedonot 117

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actuallyproposethatthemethodswepresentherearethemethodneuronsinfactuse;we useneuronsprimarilyfordemonstrationpurposes. Ifweconsidertheinputs(andprogramming)toberepresentedasatimeinstancewe canseefromFigure 6-2 (f)thatatdierentmomentsintimethissignalis\high"or\low", wecaninterpretthesetwostatesasalogical1or0attheoutput.Thereforeitisonlya matterof\when"wesamplesuchasignaltoobtainadierentresponse,i.e.dependingon whichtimeinstancewechosetosamplethesignalweobtaina1or0. Thereforethemethodutilizesthisbehaviourtoprovideaexiblelogicgatesystem byvaryingtheexact\momentofobservation"(t total )ofthesignaldependingonwhich gatewillbeperformedonwhatdata.The\exible"partshouldbeevident,aswecan easilychangethe t prog fromonecycletothenext,enablingthesystemtoperformanytwo dierentcomputationaloperations. Table6-1.Thetruthtableofeachofthevefundamentallogicgates,AND,NAND,OR, NOR,X OR. I 1 I 2 ANDNANDORNORX OR 0 00101 0 1/0 0/10110 1 1 11010 0 Purelyasareferenceexample,basedonthesignalofFigure 6-2 (f)wegiveinTable 6-2 valuesforthedierenttimestoperformeachofthevelogicgates,oneachofthe threepairsetsofinputs.Ineachofthethreecasesofinputs,thetimelengthrepresenting alogical1istwotime\units"long( t in =2).WeutilizeasignalasgiveninFigure 6-2 (f), withclockcycledenedtobe9timeunitslong.Dependingonwhichlogicoperationwe wishtoperformandalsoonwhattheinputsare,asamplinginstancewithintheclock cycleisdened,asexplainedaboveandgivenby t total .Thissamplinginstanceisintime unitswithreferencetotheinitializationfromtheclockcycle.WetaketheNORgateasan illustrativeexample.FortheNORgatewehave t prog =5timeunits.Forthesetofinputs (0,0)wesamplethesignalatthe5 th ( t prog +0+0)timeunitobtaininga1attheoutput, seeFigure 6-2 (f).Inthecasewhereourinputsare(1,0)or(0,1)wehaveasampling 118

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Figure6-2.(a)Input-OutputrelationshipofANDgate;(b)Input-Outputrelationshipof NANDgate;(c)Input-OutputrelationshipofORgate;(d)Input-Output relationshipofNORgate;(d)Input-OutputrelationshipofXORgate; (f)CombinedgureoftheInput-Outputrelationshipsofthevefundamental gates.Dierentsectionsofthisgraphrepresentdierentinstancesof Input-Outputoperations.The15dierentcases,ofthegatesandinputs considered,aremarked;with markinganOutputof1and # markingan Output0. Table6-2.Timevalues,inarbitrarytime\units"thatproduceeachofthevegates considered,basedonthesignalofFigure 6-2 (f ). I 1 I 2 ANDNANDORNORX OR 0 004253 1/0 0/126475 1 14869 7 instanceof7timeunits,givenby( t prog + t in +0),obtainingalogical0attheoutput. SimilarlyforthecaseNOR(1,1)wehave( t prog + t in + t in )givingasamplingtimeatthe9 th timeunitagainobtainingalogical0attheoutput.Thuswehaveacorrectrepresentation oftheNORgate.Usingasimilarscheme,andbysimplychangingtheprogrammingtime length, t prog wecanswitchtoperformingadierentgate. 119

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6.1.2SearchAlgorithm Promotingthisideafurther,onecannowconsidereachmomentintimeasan opportunitytorepresentanyspeciccomputationalinstructionordata.Inorderto demonstratemorepracticalapplicationsofusingtimeforcomputationalcommands(data andprogramming),wewilldemonstrateavariationonouralgorithmforperforminga searchonagivenunsorteddatabase[ 28 ]. Theconstructionofasearchenginerequiresthefollowingsteps: 1.Arepresentationofdatainthedatabase; 2.Thematchingofdatawithsearchedforitems; 3.Localizationofthematcheddata. Anygivendatabaseisconstructedoutof M uniqueitems,wealreadyreferredto thissetofitemsasthe\alphabet",andfurthereachitemcanberepresentedbyaunique naturalnumber,seeSection 3.1 .Wehaveshownabovethatanitetimeintervalcan representalogical1,wecanextendthisideatotimelengthsofmultiplesizes,eachone representingadierentnumber.Thusweutilizedierenttimelengthstorepresenteach oneoftheuniqueitemsthatconstituteagivenalphabet.Thesetimelengthscanbe markedbyadeltafunction(aspikeinneuralcontext)occurringataspecicmomentin timewithrespecttotheinitializationofaclockcycle;theintervalbetweeninitialization andthedeltafunctionisthetimelengthforaspecicitem.Sogivenanydatabasewecan representitbyaseriesofdeltafunctiongenerators, R x ,allreceivinginitializationfromthe clockcycle,andeachprovidingaspikeatthespecicdelayedmomentintimeprovidinga timeintervalwhichrepresentsthedataitemitstores,seeFigure 6-3 .AsshowninFigure 6-3 wecanconnectthesefunctiongenerators, R x ,eachindependentlytoaCoincidence DetectorUnit(CDU).ThefunctionoftheCDUsistoproducea\conrmation",aspike, atthemomentwhenitreceivestwosimultaneousspikes.The\searchelement"isanother deltafunctiongenerator, R search;k settoproduceaspikeatthemomentthatrepresents thedataitemwewishtosearchfor.Asaresultthecoincidencedetector(s)producea 120

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spikeonlyifthesearchedforitemiswithinthegivendatabasesinceonlythenwillaCDU receivetwosimultaneousspikes(fromadatabaseelementandthesearchedforitem).Note thattheterminologyweuseissuchastorelatethisabstractpresentationofthemethod directlytotheneuralimplementationtofollow,specically R willbeaneuralsynapse strengthcoecient,whiletheTDUandCDUwillbespecicneuralcircuits. Forthenalstepoflocalizationofpositiveresultsofasearchweconnectallthe coincidencedetectorstoanaccumulator,IntegratingUnit(IU),eachweightedbya dierentweightfollowingGodelnumbering[ 31 ],i.e.therstCDUweightedby2 0 ,the secondby2 1 ,thethirdby2 2 andsoon,providingusawaytouniquelyidentifythe positionsofallmatcheditems,bymeasuringthe\height"oftheweightedsumateach momentintime. Inadditionwecanperformmultiplesearcheswithinthesameclockcycle;i.e.we canconnecttotheCDUsmultiplesearchelements,eachencodingadierentitemand searchforalltheitems,technically,atthe\sametime".Performingmultiplesearchesdoes notcreateaproblemattheaccumulatorsincetheconrmationspikesfromtheCDUs willarriveatdierenttimes,fordierentitems,andthusspikesofthesameitemwillbe accumulatedseparatelyfromspikesofadierentitem. Giventhatourimplementationoflogicandapplicationtodatabasesearchingisbased onsamplingsignalsatanappropriatetimethemainlimitationofourmethodsis:\how nelycanwe`slice'time?"Fortheimplementationoflogicgates,samplingratewilldene howfastwecanperformalogicoperation,withahighersamplingratethesmallerthe widthofthesquarepulsecanbeandthustheshorterthetimeittakestocompletea clockcycle(performanoperation).Inpracticewearelimitedbythetechnologyused toimplementourmethod.Forthesearchalgorithm,thehigherthesamplingratethe moreitemswecanstorewithinagivenclockcycle.Thereisonemorelimitationtothe searchalgorithm,thefactthatwithincreasingsizeofthedatabasetheweightsforeach 121

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Figure6-3.Schematicrepresentationofthetimebasedsearchmethod. locationincreaseexponentially,thiswouldcreateaproblembasedonthetechnologyused toimplementthemethod,butcanbeaddressedbyamulti-stageaddressingsystem. 6.2NeuralImplementation OurdemonstrationswithneuralcircuitsaremainlyinspiredbytheworkofAbarbanel et.al.[ 93 94 ].SpecicallywetakeaneuralcircuitcreatedbyAbarbanelet.al.designed forinterspikeintervalrecognition,andweextractfromthiscircuittwocompartments,the TimeDelayUnit(TDU)andtheDetectionUnit(DU)(whichwerenametoCoincidence DetectinoUnit(CDU)).Weutilizethesesub-circuitstomanipulatethetimingofaction potentials,bothfortheircreationanddetection. 6.2.1NeuralModels Thisshortsectionisaslightdeviationfromourmainthemetoverybrieyintroduce twoclassicneuronmodelsandthreemodelsofsynapticconnections,whichweusefor ouralgorithmimplementations.Thisispurelydonetoplacethereaderin,atleast,some 122

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contextofneuroscience;forfurtherexpositionswedirecttoworkbyIzhikevich[ 96 { 98 ]and Lathamet.al.[ 99 100 ],amongmanyothers. TypeIneuron .ThisisamodelintheframeworkofHodgkinHuxley,asingle compartmentmodelwithfastsodiumchannel,adelayedrectierpotassiumchannelanda leakchannel[ 96 98 101 ]denedbythefollowingequation: C dV ( t ) dt = I DC + g Na m 3 ( t ) h ( t )( E Na )]TJ/F2 1 Tf0.99999 0 Td[(V ( t ))+ g K n 4 ( t )( E K )]TJ/F2 1 Tf0.99999 0 Td[(V ( t )) + g L ( E L )]TJ/F2 1 Tf0.99999 0 Td[(V ( t ))+ I S ( t ) ; (6{1) where V ( t )isthemembranepotential, I DC theexternalDCcurrent, I S ( t )thesynaptic current,andbydening r = Na;K;L torefertothesodium,potassiumandleak channels,wehave E r asthereversalpotentialsand g r astheconductanceofeachchannel. Inoursimulationswehaveusedthefollowingparametervalues: C =1 Fcm )]TJ/F5 1 Tf0.8264 0 Td[(2 E Na =50mV, E K =-95mV, E L =-64mV, g Na =215mScm )]TJ/F5 1 Tf0.8264 0 Td[(2 g K =43mScm )]TJ/F5 1 Tf0.82639 0 Td[(2 g L =0.813mScm )]TJ/F5 1 Tf0.8264 0 Td[(2 .Finallythegatingvariables X = f m;h;n g satisfythefollowing equation: dX ( t ) dt = X ( V ( t ))(1 )]TJ/F2 1 Tf0.99999 0 Td[(X ( t )) )]TJ/F2 1 Tf0.99999 0 Td[( X ( V ( t )) X ( t ),where X and X aregivenby: m = : 32(13 )]TJ/F5 1 Tf0.8264 0 Td[(( V ( t ) )]TJ/F3 1 Tf0.8264 0 Td[(Vth )) e (13 )]TJ/F12 1 Tf0.96295 0 Td[(( V ( t ) )]TJ/F15 1 Tf0.96295 0 Td[(Vth )) 4 : 0 )]TJ/F5 1 Tf0.8264 0 Td[(1 m = 0 : 28(( V ( t ) )]TJ/F3 1 Tf0.8264 0 Td[(Vth ) )]TJET0.504 w264.67 400.96 m345.82 400.96 lSBT/F5 1 Tf7.9706 0 0 7.9706 0 0 Tm41.904 50.643 Td[(40) e (( V ( t ) )]TJ/F15 1 Tf0.96296 0 Td[(Vth ) )]TJET0.36 w272.52 393.91 m330.48 393.91 lSBT/F12 1 Tf5.9779 0 0 5.9779 0 0 Tm53.583 66.2 Td[(40) 5 )]TJ/F5 1 Tf0.8264 0 Td[(1 h = : 128 e 17 )]TJ/F12 1 Tf0.96295 0 Td[(( V ( t ) )]TJ/F15 1 Tf0.96295 0 Td[(Vth ) 18 h = 4 e 40 )]TJ/F12 1 Tf0.96295 0 Td[(( V ( t ) )]TJ/F15 1 Tf0.96296 0 Td[(Vth ) 5 +1 n = 0 : 032(15 )]TJ/F5 1 Tf0.8264 0 Td[(( V ( t ) )]TJ/F3 1 Tf0.8264 0 Td[(Vth )) e (15 )]TJ/F12 1 Tf0.96295 0 Td[(( V ( t ) )]TJ/F15 1 Tf0.96295 0 Td[(Vth )) 5 )]TJ/F5 1 Tf0.8264 0 Td[(1 n = 0 : 5 e ( V ( t ) )]TJ/F15 1 Tf0.96295 0 Td[(Vth ) )]TJET0.36 w267.41 340.2 m319.61 340.2 lSBT/F12 1 Tf5.9779 0 0 5.9779 0 0 Tm52.248 57.216 Td[(10 40 where V th = )]TJ/F0 1 Tf0.77777 0 Td[(65mV.Inthismodelthefrequencyofspikingasfunction,whichinthe contextofouralgorithmdenesaclockcycle,isgivenby: f =C p I DC )]TJ/F2 1 Tf0.99999 0 Td[(I 0 ,where I 0 is thethresholdforspikingandCisaconstantbasedonthemodelparameters. Bistableneuron .AgainthismodelderivesfromHodgkinHuxleyasasingle compartmentmodelwithaleakcurrent,butthesodiumchannelispersistentandthe potassiumchannelfast,thedeningequationisgivenby: C dV ( t ) dt = I DC + g Na m 1 ( V ( t ))( E Na )]TJ/F2 1 Tf0.99999 0 Td[(V ( t ))+ g K n ( t )( E K )]TJ/F2 1 Tf0.99999 0 Td[(V ( t )) + g L ( E L )]TJ/F2 1 Tf0.99999 0 Td[(V ( t ))++ I S ( t ) ; (6{2) 123

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wherethegatingvariable n ( t )isgivenby: dn ( t ) dt = n 1 ( V ( t )) )]TJ/F3 1 Tf0.8264 0 Td[(n ( t ) n and f m 1 ( V ) ;n 1 ( V ) g = X 1 ( V ) bytheactivationfunction: X 1 ( V )=1 = (1+exp(( V X )]TJ/F2 1 Tf1.0212 0 Td[(V ) =k X ));theremainingvariables arethesameasinthetypeIneuronmodel,seeEquation 6{1 .Inoursimulationswe haveusedthefollowingparametervalues: C =1 Fcm )]TJ/F5 1 Tf0.8264 0 Td[(2 E Na =60mV, E K =-90mV, E L =-80mV, g Na =20mScm )]TJ/F5 1 Tf0.8264 0 Td[(2 g K =10mScm )]TJ/F5 1 Tf0.82639 0 Td[(2 g L =8mScm )]TJ/F5 1 Tf0.8264 0 Td[(2 V m =-20mV, V n =-25mV, k m =15mV, k n =5mV, n =0.16mV.Thismodelmainlydiersfrom thetypeIneuron,asitsrestingstateandoscillatingstatecanbothbestableandasa resultthesystemcanswitchfromonestatetotheothersimplybyavariationinitsstate variable( V ( t )),unliketypeIforwhich,forbehaviouralchangesasystemparameterneeds tobevaried( I DC )[ 96 98 ];actuallyweutilizethisdierenceinbehaviourtoconstructthe TDU. Synapses .Weutilizeinourneuralcircuitsthreetypesofsynapticconnectionsand theircurrentsaregivenby: ExcitatoryAMPA: I S ( t )= g A S E ( t )( E E )]TJ/F2 1 Tf1.1245 0 Td[(V ( t )),where g A isthesynaptic conductance, E E =0mVisthereversalpotentialand S E ( t )thegatingvariable,that isbiologicallyrelatedtothefractionofboundglutamatereceptors; ExcitatoryNMDA: I S ( t )= g N B ( V ( t )) S E ( t )( E E )]TJ/F2 1 Tf1.0346 0 Td[(V ( t )),where g N isthesynaptic conductanceand B ( V )=1 : 0 = (1+0 : 288exp( )]TJ/F0 1 Tf0.77777 0 Td[(0 : 062 V )). E E and S E ( t )arethesame quantitiesasfortheAMPAsynapse; InhibitoryGABA A : I S ( t )= g G S I ( t )( E I )]TJ/F2 1 Tf1.1227 0 Td[(V ( t )),where g G isthesynaptic conductance, E I = )]TJ/F0 1 Tf0.77777 0 Td[(75mVisthereversalpotentialand S I ( t )thegatingvariable, relatedtothefractionofboundgabareceptors. Thegatingvariablesaregivenby: S X ( t )= S 0 ( ( t )) )]TJ/F3 1 Tf0.8264 0 Td[(S X ( t ) ^ ( S 1 )]TJ/F3 1 Tf0.8264 0 Td[(S 0 ( ( t ))) ,where S X ( t )= f S E ( t ) ;S I ( t ) g S 0 ( )=0 : 5(1+tanh(120( )]TJ/F0 1 Tf0.83446 0 Td[(0 : 1)))isasigmoidalfunction,andgiventheheavisidefunction ( X )=1if X> 0,else( X )=0wehave ( t )= P i ( t )]TJ/F2 1 Tf0.91874 0 Td[(t i ) (( t i + R ) )]TJ/F2 1 Tf0.91874 0 Td[(t ) ( X ),here t i isthetimeofthe i th presynapticneuronalspike.Finallywehavetwotimeconstants R =^ ( S 1 )]TJ/F0 1 Tf1.0062 0 Td[(1)and D =^ S 1 ,inbiologicalsensethetimefortheneurotransmittertobind andrelease,respectively;fortheAMPAsynapse R =0 : 1msand D =1 : 5ms,forNMDA R =2 : 5msand D =70ms,andforGABA A R =1 : 12msand D =5 : 5ms. 124

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TimeDelayUnit(TDU)andCoincidenceDetectionUnit(CDU) .TheTime DelayUnitisconstructedwithoneofeachofthetwoneuronmodelswepresented,a typeIneuronandabistableneuron,ourconstructionfollowsdirectionsgivenin[ 93 ].A schematicofthecircuitisgivenintheinset(a)ofFigure 6-4 .Neuron istypeIneuron andneuron isabistableneuronandbothneuronsareinitiatedatarestingpotential state.TheinputconnectionstotheneuronsareaAMPAsynapsetoneuron anda NMDAsynapsetoneuron .Theneuronsarealsoconnectedtoeachotherthrough aGABA A synapse, )324()]TJ/F9 1 Tf0.78306 0 Td[( ,andaNMDAsynapse, ,itisthissynapsethatwe modulatetocreateadelay.Aswehavethecircuitconguredanincomingactionpotential arrivingat t 0 isdirectedintobothneurons.Forneuron ,sinceitisatypeIneuron, asingleincomingactionpotentialhasnoeectotherthantoincreasethemembrane potentialslightly,buttherewillbenobehaviouralchange.Forneuron thoughthe situationisdierent,asitisabistableneuronanincomingactionpotential\pushes" theneurontoitsoscillatingstate,soneuron startsgeneratingactionpotentials.These actionpotentialsdrive,throughtheconnectingNMDAsynapse,neuron andeventually will\push"themembranepotentialofneuron enoughforittoreitsown\output" actionpotential.Thetimingofthisoutputpotentialdependsonhowlongittakesfor neuron topushneuron overitsringthreshold,andofcoursethistimedelaycanbe modulatedbya\stronger"or\weaker"connectionbetweenthetwoneurons.Specically thestrengthoftheaforementionedNMDAsynapse, ,canbevariedusingaweight parameter R as g N R = R g N ,with g N =1mScm )]TJ/F5 1 Tf0.8264 0 Td[(2 .Thereforeanincomingaction potentialattime t 0 willbetechnically\absorbed"bythesystemandanoutputaction potentialwillbegeneratedatalatertime t 0 + ( R );inessencethough,sincethereis nodierencebetweenthetwoactionpotentials,ineectwhatweachieveisthedelayin thepropagationofasingleactionpotential.Throughsimulationswehavequantiedthe eectof R on ( R )andtheirrelationshipispresentedinFigure 6-4 .Ofcourseweneed tomakesurethatonlyasingleoutputpotentialiscreated,thisisachievedthroughthe 125

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Figure6-4.TimeDelayUnit(TDU).Therelationshipbetweentheparameter R ,which denessynapticstrength,andthetimedelaycreated .Inset(a):Aschematic representationoftheTDUneuralcircuit.Inset(b):Domainandrangewhere R vs. ( R )islinear,andhencetherangeof R valuesusedforexiblelogic gateimplementation.(Adaptedfrom[ 93 ].) inhibitoryGABA A synapsebetweenthetwoneurons, )260()]TJ/F9 1 Tf0.95218 0 Td[( ,whichtakesthe\output" spikeofneuron anddirectsit,besidestotheoutput,backintoneuron inhibitingit andpushingitawayfromitsoscillatingstatebacktoitsrestingstate,ceasingfurther generationofactionpotentialsfromneuron ,andasadirectconsequencepreventsthe generationoffurtheractionpotentialsfromneuron itself. TheCoincidenceDetectionUnitisamuchsimplercase,itissimplyasingletypeI neuronthatactsasaclassicintegratorreceivinginputsfromtwosourcesthroughtwo AMPAsynapses.Thestrengthoftheconductanceofthetwosynapsesisequatedso thatbothinputsourceshaveequaleectontheneuron.Inadditionthestrengthisset highenoughfortheneuronnottocrossitsringthreshold,unlessitreceivestwospikes 126

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simultaneously(orwithintoleranceofasmalltimeinterval,denedbytheconductanceof thesynapses). 6.2.2AlgorithmImplementations Therstneuralbasedimplementationoflogicemulationissimplybasedontheidea ofsamplingthemembranevoltagesignalofasingleneuronatdierentmomentsintime. Thesecondmethodutilizesacircuitoftwobi-directionallycoupledHodgkinHuxley neurons,theTDUcircuit.Logicgatesareemulatedbyvaryingthesynapticstrength R thatdeterminesthetimedelay ( R )andthenobservingtheoutputatapredetermined xedtimeinstant.Thesearchalgorithmisstraightforwardlyimplementedalongthe directionsgiveninthemoregeneralcontextpresentationofSection 6.1.2 Varyingthe\momentofobservation" .Thisistrulyaverysimpleimplementation. Wejustconsiderasingleneuronproducinganactionpotential,seeinsetofFigure 6-5 Sincethissignalisaproductofsimulationweconsidereachpointgeneratedbythe simulationasaninstanceintimeandwiththezerothsimulationpointasareferencepoint wedene t total accordingtothecomputationaloperationtobeperformed,exactlyasin Section 6.1.1 TaketheNORgateasaspecicexample.InFigure 6-5 weshowtheresultsof samplingoursignalatthetimeinstancesof2555fortheoperationNOR(0,0),2605for NOR(0,1)/(1,0)andat2655forNOR(1,1).Intermsofprogramming( t prog )andinput timelengths( t in )wehave t prog =2555timesteps,specictotheNORgate,and t in =50 timesteps.Asmalldeviationfromthegeneralcaseisthattheactionpotentialactually uctuatesbetween48mVand-92mV,itisnotsimply0or1.Thereforewesetavoltage threshold,at-45mV,overwhichwedenethelogical1andunderwhichthelogical0,at theoutput. AsisclearfromFigure 6-5 theappropriateOutputvaluesareobtained,thatisthe voltageisover-45mVforthe(0,0)caseandbelowfortheothertwocases.Thewhole procedurecanbeinterpretedasfollows:toperformthelogicgateNORwerequireto 127

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Figure6-5.DemonstrationofoperatingaNORgate,withasingleneuronusingdierent samplingtimes.Dependingontheinputsadierentsamplingmomentis denedbeyondthe2555 th momentthatdenestheNORgate.Eachpossible caseisshownwithacircle,blackcircle( )indicatinga1attheOutput, emptycircle( # )a0.TheOutputisdenedbyavoltagethresholdat-45mV, shown.Inset:Thewholeactionpotentialconsidered. Table6-3.Appropriatetimesampleinstances,basedonsimulationpoints,toperform eachofthevegatesconsidered.Inbracketsthevoltageoftheactionpotential, ateachpoint,isgiv en. I 1 I 2 ANDNANDORNORX OR 0 02405(-57mV)2505(-9mV)2455(-56mV)2555(1mV)2485(-52mV) 1/0 0/12455(-56mV)2555(1mV)2505(-9mV)2605(-86mV)2535(34mV) 1 12505(-9mV)2605(-86mV)2555(1mV)2655(-86mV)2585(-50 mV) wait2555timestepsasaprogrammingtimelengthandthenanother50timestepsfor eachoccurrenceofasinglelogicalinputof1.Usingthesamesystem,thatisasingle HodgkinHuxleyneuronproducinganactionpotentialatthesamerate,timeinstances forreadinganOutput,withrespecttothezerothsimulationpoint,alongwiththeir resultingrespectivevoltagevalue,foraccomplishingthevefundamentalgatesaregiven inTable 6-3 .Forconrmationwereferbacktothegatetruthtable,Table 6-1 128

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Varyingthemomentofgeneration .Conversetothepreviousimplementation, inthisimplementationwexthemomentofobservation,inthespecicexampleat t =425msfromaninitiatingactionpotentialat t 0 ,andwevarythemomentofactual generationofthesignal.Weuse\actualtime"insteadofsimulationpointstoemphasise theindependencebetweenthemomentofobservationandthecomputationaloperation. Tohavetheabilitytogenerateanactionpotentialatdierenttimes,withrespectto t ,weuseaTDUaspartofourcomputationalsystem.FollowingFigure 6-4 ,theTDU receivesaninputactionpotentialat t 0 andproducesadelayedoutputactionpotential at t 0 + ( R ),itisthemomentofgenerationofthisoutputactionpotentialweusefor computation.Exactlyasexplainedbefore,withrespecttotimeinstances,weutilize R valuestakenfromtherelationship ( R ),andwith R total = R prog + R I 1 + R I 2 werepresent adesiredcomputationaloperation.Notethatweusethedomain1 : 68
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Figure6-6.DemonstrationofoperatingaNORgate,withaneuralcircuitusingdierent delaytimes.Allthreecasesofinputsareshown,inactualityonlyoneaction potentialisgeneratedbythesystem.Thetimeofgenerationoftheaction potentialisinverselyproportionalto R .Giventhe\momentofobservation"at 425ms,theOutputisobtainedusingthevoltagethresholdat-45mV.Once again is1attheOutputand # 0. actionpotential,whichresultsinavoltageof-16mVatthemomentofobservationof 425ms,whichisoverthethresholdof-45mV,thustheoutputis1.Forthecases(1,0) and(0,1),whicharedegenerate,wehave R total =1 : 705+0 : 005+0.Theactionpotentialis generatedwithapeakat424ms,sobythetimeofthemomentofobservationthevoltage drops,belowthethreshold,to-89mV,producinga0attheoutput.Wehaveasimilar resultforthethirdandnalcaseof(1,1)attheinputs, R total =1 : 705+0 : 005+0 : 005, thepeakoftheactionpotentialhappensat423.5ms,soagainthevoltagedropsbelow thethresholdbythetimeofthemomentofobservation.Itisclearthatoursystemworks perfectlytoreproducethelogicgateNOR,similarresultshavebeenobtainedfortheother ofthefourfundamentalgates,the R values,andtherespectivevoltagesatthemomentof observation,aregiveninTable 6-4 ;whichonceagaincanbecomparedwithTable 6-1 130

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Table6-4.Appropriate R valuessotimeshiftanactionpotentialinordertoperformeach ofthevegatesconsidered.Theactionpotentialissampledalwaysatthesame momentintime,425msafteraninitiatingactionpotential.Inbracketsthe voltageoftheactionpotential,at425ms,isgiv en. I 1 I 2 ANDNANDORNORX OR 0 01.690(-58mV)1.700(48mV)1.695(-56mV)1.705(-16mV)1.697(-53mV) 1/0 1/01.695(-56mV)1.705(-16mV)1.700(48mV)1.710(-89mV)1.702(35mV) 1 11.700(48mV)1.710(-89mV)1.705(-16mV)1.715(-85mV)1.707(-53 mV) Neuralsearchalgorithm .Thesearchmethodcanalsobeimplementedwiththe sameneuronalcircuits,TDUandCDUalongwithanIntegratingUnit(IU),simply asingleneuronwithitsthresholdforringanactionpotentialsetveryhigh.The rstcomponentofoursearchengineisencodingbyeachuniquemomentintimea databaseelement.Aswehaveshownabove,itemsfromanygivenalphabetcanbe representedbytimeintervals.Wedothesame,butinsteadofusingactualtimeintervals forrepresentationofitemsweuseunique R values.Soadatabaseofsize N ,regardlessof size,canbeconstructedusing N TDUseachonerepresentingoneofthe M distinctitems ofthealphabetassociatedwithaunique R valueandasaresultauniqueinstanceintime forgeneratinganactionpotential. Thesecondcomponent,justasintheabstractedexplanationabove,isaCoincidence DetectorUnit(CDU),thisisasingleneuronatrestingpotential.Thepropertyofthis neuronthatmakesoursearchengineworkisthatitrequirestwoactionpotentialsto arriveatitsmembranesimultaneouslyfortheneurontoproducearesponseaction potential.Ifandwhen,twoactionpotentialscoincidetheCDUwillgenerateitsown actionpotential,otherwiseitwillnotre,providingusawaytoconrmtheexistenceor notofasearchedforiteminthedatabase. SonowwecanincorporateinourmodeltheSearchEngineelements,TDUssetatthe R valuesfortheitemswewishtosearchfor.WecoupleeachCDUwitheachTDUofthe databaseandtoalltheSearchEngineelements,referringbacktoFigure 6-3 ,sothatwhen theSearchEngineelementsre,theCDUsthatareattachedtothesearchedforitemswill 131

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receivetwosimultaneousactionpotentialsandrethemselves.Ifasearchedforitemdoes notexistinthedatabasenoCDUwillreceivetwosimultaneousspikessotherewillbeno actionpotentialfromtheCDU. FinallyweconnectallCDUstoanIntergradingUnit(IU)tocombineallsearch results.EachCDUisconnectedtotheIUwithasynapseofdierentconductanceweight ( g ),followingGodelnumbering;therstCDUisconnectedwithaweight2 0 g thesecond withweight2 1 g ,thirdunitwith2 2 g andsoon.Thisprovidesuswithawaytouniquely markeachpositionofeachiteminthedatabase.Attheoutputwereadthevoltageofthe IUandinterpretthesearchedforresults. Therearetwoparameterstoconsideratthereadingoftheresult,rstisthetime instanceatwhichavoltageincreaseoccurs,whichrepresentsthesearchedforitem,and secondistheamountbywhichthevoltageoftheIUisincreased.Basically,eachmoment intimerepresentsaninformationitem,asdenedfromthedierent R valuesweuseto encodethem,andeachincreaseinvoltagerepresentsapositioninthedatabasegivenby theGodelnumbering. 6.3ElectronicImplementation Asitturnsouttheelectronicimplementationofthismethodcanbeperformedwith simplemultipliersinsteadofacomplicatedsamplingcircuitofvariabledelays.Basically wegeneratetwosignals,onerepresentingthe\genericsignal",asinSection 6.1.1 ,and theothersignalisgeneratedatthetimeinstancethatrepresentsthecomputational operationtobeperformed,whetherlogicoperationorsearchoperation.Themultiplier simplymultipliesthetwosignalsandtheproductofcoursewillbe1onlyifthetwosignals coincideintime,inessenceweareemulatingtheneuralCoincidenceDetectionUnit. Logicgatesalgorithm .AsshowninFigure 6-7 weconnectthereferencesignal squarepulsegenerator(Vr)toamultiplier(U1)alongwiththevarying\sampling signal"generator(Vs).Thevaryingsamplingsignalgeneratoremulatesthemoment ofobservationbyproducingashortsquarepulsesignalwithvaryingdelaycomparedto 132

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Figure6-7.Schematicrepresentationofanelectroniccircuitforlogicusingtime.Vr providesthereferencesignalwhileVsthe\momentofobservation".Thetwo signalsswitchfrom1Vto0V,thereforeU1throughsimplemultiplication providestheOutput,givenstraightforwardlybythevoltageofresistorR1(1V logical1,0Vlogical0). thereferencesquarepulseofVr.ThefunctionofthemultiplierU1istosimplymultiply thetwosignalsandthereforetheresultingsignalisthesameasifsamplingthereference squarepulseVratthemomentofobservation. Forourimplementationweusedareferencesignaloffrequency10Hzandpulsewidth 30ms,showninFigure 6-8 (a),asaresultourtimelengthrepresentingalogical1isof 30ms(=t in ).The\samplingsignal"fromVsisagainapulseoffrequency10Hz,butits generationinstancecanbevariedcreatingavariabledelaysignal,seeFigures 6-8 (b)(d)(f). Thetimedelayofgenerationofthe\samplingsignal"isourprogrammingtimelengthplus theappropriatedelayforeachofthethreepairsofinputs,valuesforprogrammingtheve fundamentalgatesandeachcaseofinputsaregiveninTable 6-5 133

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Figure6-8.DemonstrationofoperatingaNORgateusinganelectroniccircuitutilizing timedependantcomputation.(a)Referencesignal,tobesampledatdierent times.ReferbacktoFigure 6-2 (f).(b)Samplingsignalforthecaseofinputs (0,0)toaNORgate,equivalentto t prog +0+0=60ms.(c)Outputsignalfor performingNOR(0,0),outputisalogical1,signalis\high".(d)Sampling signalforthecasesofinputs(1,0)/(0,1)toaNORgate,equivalentto t prog + t in +0=90ms.(e)OutputsignalforperformingNOR(1,0)or NOR(0,1),outputisalogical0,signalis\low".(f)Samplingsignalforthe caseofinputs(1,1)toaNORgate,equivalentto t prog + t in + t in =120ms. (g)OutputsignalforperformingNOR(1,1),outputisalogical0. Table6-5.DelaytimesforVs,themomentofthe\samplingsignal"withrespecttothe startoftheclockcycle,toimplementeachofthevegates considered. I 1 I 2 ANDNANDORNORX OR 0 00ms40ms10ms60ms30ms 1/0 0/130ms70ms40ms90ms60ms 1 160ms100ms70ms120ms90 ms 134

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AsaspecicillustrativeexampletaketheNORgate.AsindicatedinTable 6-5 theprogrammingtimelengthfortheNORgateis60ms,andasstatedabovethetime lengthrepresentingalogical1attheinputsis30ms.Thereforeforthesetofinputs (0,0), t prog +0+0,wegeneratea\samplingsignal"withsimplyadelayof60ms,see Figure 6-8 (b).Forthesetsofinputs(1,0)and(0,1), t prog + t in +0,wegeneratea\sampling signal"withdelay90ms,seeFigure 6-8 (d);andnallyfortheset(1,1), t prog + t in + t in ,we generatea\samplingsignal"withdelay120ms,seeFigure 6-8 (f).Theoutputisreadat theresistorR1andisshowninFigures 6-8 (c)(e)(g),asVOut.FromFigures 6-8 (c)(e)(g) wecanseethatthecorrectOutputisgeneratedforeachcaseofinputsforreproducingthe truthtableofaNORgate. Searchalgorithm .Forthesearchmethod,againourelectronicimplementationis greatlysimpliedwiththeuseofmultipliers,U1,U2,U3,simplyinplaceofCoincidence DetectionUnits,seeFigure 6-9 forthespecicschematic.Asaspecicexampleconsider encodingthenumbers f 1 ; 2 ; 3 g inthreedatabaseelementsR i ,R j ,R k ,using m t ,where t =10msand m 2 N ,representinganyabstractitem.InthesearchelementsRS1, RS2,RS3,wehaveencodedoursearchfornumbers,specicallyallthree f 1 ; 2 ; 3 g .The theaccumulatedsignalfromthesearchelementsisfedindependentlytoeachmultiplier, alongwiththesignalofeachmultiplier`sassociateddatabaseelement.Theresultfrom eachmultiplierisasignalofasquarepulseifthesearchedforitemexistsorazerosignal ifitdoesnotexist.Forthenalstepeachmultiplierisconnectedtoanaccumulator sub-circuitthroughresistors,R7,R8,R9whoseactualresistancefollowsGodelnumbering. Thereadoutissetattheoutputoftheaccumulatorthatcollectsthesignalsfromthe multipliers,atU5. AsseeninFigure 6-10 (b)allthreeitemsareidentiedasexistinginthegiven database,thetimelyspikesattheoutput,andtheirlocationinthedatabaseisgivenby thevoltageheightoftheirrespectivespike.SpecicallyweseeinFigure 6-10 (b)three spikes,therstspikeat10msrepresentstheoccurrenceofitemgiventhenumber1in 135

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Figure6-9.Schematicrepresentationofanelectroniccircuitforthetimedependant searchmethod.TheR-unitsaresignalgenerators,Rxbeingthegenerators usedforencodingthedatabase,RSibeingthegeneratorsusedforencoding searchedforitems.TheU-unitsaremultipliers,replacingtheCDUsofFigure 6-3 .R7,R8,R9aretheresistorsassociatedwitheachdatabaseelement deningitslocationthroughtheirresistancevalue.Thedash-boxed sub-circuitsareaccumulatorcircuits(relatingtoIUs)\puttingtogther"botha \searchsignal",inthecaseofmultiplesimultaneoussearches,andtheresult signalfromthemultipliers. thedatabaseandtheheightof1Vrepresentsthefactthatthisitemisstoredinthe rstlocationofthedatabase.Analogouslyfortheothertwospikesrepresentingitems numbered2and3,attimeinstancesof20msand30msrespectively,againtheheight ofeachspikesigniesthelocationoftheitem;2Vfor2 nd positionand4Vforthe3 rd positioninthedatabase. 6.4Discussion Wehavepresentedtwoalgorithmsthatusetimeintervalsforcomputation;therst forperformingexiblelogicandthesecondforsearchingaunsorteddatabase.Thecentral 136

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Figure6-10.Demonstrationofperformingasearchusinganelectroniccircuitutilizing timedependantcomputation.RefereingtotheschematicofFigure 6-9 :(a) TheoutputofthesearchkeyaccumulatoratU6,thisisthe\searchsignal" fromallthesearchelementsthatisfedtoallmultipliers;(b)Theoutputof theresultaccumulatorU5,conrmingtheexistenceofitemsitems f 1,2,3 g withpulsesat10ms,20ms,and30ms,respectively;alsoprovidingthe locationofeachiteminthedatabase,throughthevoltageofeachpulse,1V, 2V,4V,respectively. ideaoftheproposedalgorithmsisthateach,andany,momentintimecanbeutilized torepresentacomputationaloperationand/ordata.Withtheuseofadenedclock cyclelengthwesegmenttimeintomanageablepieces,withinwhichwedenemomentsin timethatwillproducedierentresponsesbasedonavariablequantity.Inourproposed algorithmstheresponsesareoftwotypes,eitherthecrossingofathresholdorthe coincidenceofevents,bothofwhichcanbeinterpretedascomputationaloperations. Withcarefulconstructionofasignal,whichencapsulatesthevefundamentallogic gates,wehaveshownhowtimeintervalscanbeusedtorepresenttheprogrammingofa universalexiblelogicgate.Inadditionwehaveshownhowazerolengthtimeinterval canrepresentalogical0andanitelengthtimeintervalalogical1.Astraightforward 137

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combination(summation)oftheprogrammingtimelengthandthelogicalinputstime lengthcanprovideamomentintimeatwhichsamplingthesignalwillproducethedesired output.Wecanthusvisualizeanarrayofmanycomputationalelementseachreceivinga commonsignal.Eachoftheseelementsisprogrammedtosamplethesignalatdierent timesproducing,inparallel,theresultsofdierentlogicaloperations.Actuallythereis noreasontoconneourselvestothe\simple"signalpresentedhere,Figure 6-2 (f).The methodcanbegeneralizedtosamplingmore\complicated"signals,reproducingeven morecomplicatedoperationsatdierenttimes;inthespiritofChapter 5 Furthermore,wehaveshownhowmanydierenttimeintervalscanbeutilizedinan algorithmtoperformsearcheswithinagivendatabase.Wehaveshownhoweachnatural numbercanberepresentedbyatimeinterval,denedbetweenthe\tick"ofauniversal clock(clockcycle)andadeltafunction.Withinoneclockcyclewesearchthedatabaseby lookingforcoincidencesbetweenoursearchforelementsandtheelementsencodedinthe database.Itissignicanttonotethatourmethodcompletesasearchwithinonesingle clockcycleregardlessofthesizeofthedatabase;anditisnotconnedtosearchingfor singleitemssequentially,butmultiplesearchescanbeperformedwithinthesameclock cycle. Forbothalgorithmspresentedweprovidedrealizations,bothwithneuraland electroniccircuits.Aswehaveshown,theimplementationsarestraightforward,especially inthecaseofelectroniccircuits.Thepresentedmethodsareadirectconsequenceof consideringdelayedtimeresponsesascomputationalcommands.Forbothimplementations themainlimitationisthesmallesttimescaleofthesystem;aconsequenceofconsidering timeascomputationaldataandprogramming,thesmallerthetimeintervalwecanutilize, themoredataandoperationscanberepresented. 138

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CHAPTER7 CONCLUSION ChaoticComputationhascomealongwaysinceitsrststepin1998[ 25 ].Both withinthe\original"groupofDittoet.al.[ 102 103 ],andevennowdayswithinthe morethemoreextendedgroupofcollaborators[ 104 105 ],theoreticalandexperimental advancementshaveestablishedChaoticComputationasarealisticpossibilityforthe futureofcomputation.Wearegladtoreportthattheeldhasalsoattractedthe attentionofresearchersbeyondourgroup[ 106 107 ]. Evenfurther,theaimofbringingresearchoutofthelaboratoryandintothepublic domainisactivelypursued;ChaologixInc.(www.chaologix.com)isanactivecompany withthesoleaimtorealizethetechnologyandpresentthesolutiontothepublic.In accomplishingthisgoalthemainattemptscirclearoundconventionalCMOSbasedVLSI circuitry[ 108 ],butasweemphasisedUniversalityallowsustoconsiderevenmoreesoteric approachessuchasmagneto-basedcircuitry[ 109 ],orevenhighspeedchaoticphotonic integratedcircuits[ 110 ]. Theaimsofthisdissertationwerethreefold.Obviouslyforemost,exposethereader tothewholeeldofChaoticComputation;inadditiontopresentthematerialwithits naturalmultidisciplinarynature,fromtheunderliningmathematics,tothephysicsthat governsoursystems,theinspirationfrombiologyandtheengineeredimplementations; noteourattemptwastoexposethereadertodetailsfromeacheld,butatthesame timewithoutdeviatingfromthemainaim.Finallyandofequalimportance,weaimedat establishingwherethe\cuttingedge"ofresearchoftheeldis,preparethegroundwork forourfuturework. Inclosingwewishtopointoutprojectionsofthefuturepathsemanatingfrom eachchapter.Developingfurthertheconnectionsimpliedinchapteroneisessential indeterminingtheultimatelimitationsofourresearchand\bootstraps"directly towherechaptertwoleftof;specicallythefurtherextensionoftheDeutsch-Jozsa 139

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algorithm,inthespiritofhowquantumcomputingprogressed,intoalgorithmsformore specicfunctions,likeFourierTransformandintegerfactorization,andevenfurther. Theimportanceofsearching,matching,patternrecognitioninourcurrentgeneral technologicalaims,drivesustoconsiderpromotingfurtherourmethodforsearching; asortingalgorithmshouldprobablybethemostimmediatetarget,withissuesof multidimensionaldata,andperhapsextrapolatedaction,ordecision,takingnotthe toofaroffuture.Whilealltheaforementionedcouldbeconsideredmainlytheoretical milestones,theimplementationsidecannotbeignored.Faster,moreecient,smaller, weshareallgoalsthatcurrentestablishedcomputerhardwaretechnologyhas,simpler electronic,orevenalternative,realizationswillbepursued.Finally,andinsomesense comingfullcirclebacktothefundamentalmathematics,throughchaptersveandsixwe havesetuptheeldforinvestigatingfurtherhowtoobtainindependencefromactualexact statevalues,and/oreventheutilizationofonlyspecicphysicalquantities,anattempt perhapstomakemoretangibletheabstractconceptof\computation". 140

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BIOGRAPHICALSKETCH AbrahamMiliotiswasborninAnnapolis,Maryland,USA,fromparentsofGreek-Cypriot descent.HewasraisedinCypruswhereheattendedTheEnglishSchoolandgained admissiontoImperialCollege,Londonin1997.In2000hegraduatedfromImperial CollegewithaBachelorofScienceinPhysicswithTheoreticalPhysics.Underthe supervisionofprofessorDr.DimitriVvedensky,hecompletedhisnalyearthesis, \Stockmarketcrashestreatedascriticalphenomena".Dr.D.Vvedenskyalsointroduced AbrahamtotheworkofprofessorDr.WilliamDittoonusingdynamicalsystemsfor computation,whomhejoinedatGeorgiaInstituteofTechnology,Atlanta,Georgiain 2000,asagraduatestudentinthePhysicsdepartment.Between2001and2003though, Abrahampausedhisstudiestopursuesomeentrepreneurialopportunities.Hemoved backtoCypruswhere,amongotherprojects,hewasinvolvedinthecreationofMegaland ComputersLTD,anITretailsalesandsmall-oce/home-ocesupportcompany,andalso thesettingupofGame-SpotInternet/GamingCafes. In2003herejoinedDr.WilliamDittoattheBiomedicalEngineeringdepartment ofUniversityofFlorida,wherehecompletedhisPh.D.inMayof2009on\Chaotic Computation". 147