Algorithmic randomness: progress and prospects PDF
Preview Algorithmic randomness: progress and prospects
AlgorithmicRandomness Thelasttwodecadeshaveseenawaveofexcitingnewdevelopmentsinthetheoryof algorithmicrandomnessanditsapplicationstootherareasofmathematics.This volumesurveysmuchoftherecentworkthathasnotbeenincludedinpublished volumesuntilnow.Itcontainsarangeofarticlesonalgorithmicrandomnessandits interactionswithcloselyrelatedtopicssuchascomputabilitytheoryand computationalcomplexity,aswellaswiderapplicationsinareasofmathematics includinganalysis,probability,andergodictheory.Inadditiontobeingan indispensablereferenceforresearchersinalgorithmicrandomness,theunifiedviewof thetheorypresentedheremakesthisanexcellententrypointforgraduatestudentsand othernewcomerstothefield. Johanna N. Y. Franklin isanAssociateProfessoratHofstraUniversityin Hempstead,NY.SheearnedherPh.D.fromtheGroupinLogicandtheMethodology ofScienceattheUniversityofCalifornia,Berkeleyandhasheldpostdoctoral positionsinSingapore,Canada,andtheUnitedStates. Christopher P. Porter isanAssistantProfessorofMathematicsatDrake UniversityinDesMoines,IA.AfterreceivinghisPh.D.inthejointprogramin mathematicsandphilosophyattheUniversityofNotreDame,Portercompleted postdoctoralpositionsattheUniversityofParis7andtheUniversityofFlorida. LECTURE NOTES IN LOGIC A PublicationofTheAssociation forSymbolicLogic Thisseriesservesresearchers,teachers,andstudentsinthefieldofsymbolic logic,broadlyinterpreted.Theaimoftheseriesistobringpublicationstothe logiccommunitywiththeleastpossibledelayandtoproviderapid disseminationofthelatestresearch.Scientificqualityistheoverriding criterionbywhichsubmissionsareevaluated. EditorialBoard ZoeChatzidakis DMA,EcoleNormaleSupe´rieure,Paris PeterCholak,ManagingEditor DepartmentofMathematics,UniversityofNotreDame,Indiana LeonHorsten SchoolofArts,UniversityofBristol PaulLarson DepartmentofMathematics,MiamiUniversity PauloOliva SchoolofElectronicEngineeringandComputerScience,QueenMary UniversityofLondon MartinOtto DepartmentofMathematics,TechnischeUniversita¨tDarmstadt,Germany SlawomirSolecki DepartmentofMathematics,CornellUniversity,NewYork Moreinformation,includingalistofthebooksintheseries,canbefoundat www.aslonline.org/lecture-notes-in-logic/ LECTURE NOTES IN LOGIC 50 Algorithmic Randomness Progress and Prospects Editedby JOHANNA N. Y. FRANKLIN HofstraUniversity,NewYork CHRISTOPHER P. PORTER DrakeUniversity,Iowa association for symbolic logic UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre, NewDelhi–110025,India 79AnsonRoad,#06–04/06,Singapore079906 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108478984 DOI:10.1017/9781108781718 AssociationforSymbolicLogic RichardA.Shore,Publisher DepartmentofMathematics,CornellUniversity,Ithaca,NY14853 http://aslonline.org ©AssociationforSymbolicLogic2020 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2020 PrintedintheUnitedKingdombyTJInternationalLtd,PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-108-47898-4Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. CONTENTS Preface............................................................. vii JohannaN.Y.FranklinandChristopherP.Porter Keydevelopmentsinalgorithmicrandomness...................... 1 HenryTowsner Algorithmicrandomnessinergodictheory......................... 40 JasonRute Algorithmicrandomnessandconstructive/computablemeasure theory............................................................ 58 MathieuHoyrup Algorithmicrandomnessandlayerwisecomputability.............. 115 JohannaN.Y.Franklin Relativizationinrandomness...................................... 134 GeorgeBarmpalias AspectsofChaitin’sOmega....................................... 175 ChristopherP.Porter Biasedalgorithmicrandomness.................................... 206 BenoitMonin Higherrandomness............................................... 232 DonaldM.Stull Resourceboundedrandomnessanditsapplications................. 301 Index............................................................... 349 v PREFACE Sincetheearly2000s,therehasbeenaproliferationofresearchinalgorithmic randomness,culminatinginthepublicationoftwotextbooks: Computability andRandomness,writtenbyAndre´Niesandpublishedin2009,andAlgorithmic RandomnessandComplexity,writtenbyRodneyDowneyandDenisHirschfeldt andpublishedin2010. Thesevolumesnotonlyexhaustivelycoveredthekey developmentsinthefieldfromthemiddleofthe1960sthroughthelate1990s butalsodetailedmanyofthecentraldevelopmentsfromthefirstdecadeofthe 2000s. Since the publication of these two volumes, research in algorithmic ran- domnesshascontinuedunabated,withaparticularemphasisonapplications of algorithmic randomness to other parts of mathematics in such areas as classicalanalysis,ergodictheory,andprobabilitytheory. Inadditiontothese developmentsinwhatonemightcall“appliedalgorithmicrandomness,”there havebeenkeydevelopmentsoverthissameperiodoftimeintwoareasthat werealreadyveryactiveintheearly2000s,namely,approachestorelativization inrandomnessandinteractionsbetweenalgorithmicrandomnessandleft-c.e. realnumbers. Lastly,therehasbeenaconsiderableamountofnewworkon notionsofhigherrandomness,i.e.,notionsofrandomnessdefinedinterms ofhighercomputability,aswellasresource-boundedrandomnesspublished duringthistime. The need for a single source detailing these many developments is thus apparent. Thegoalofthepresentvolumeistofillthisgapintheliterature. Theeditorsofthisvolumehavecommissionedsixsurveysbyvariousexperts inthefieldandcontributedthreesurveysoftheirown. Althoughnarrowerin scopethanboththeNiesandDowney/Hirschfeldtvolumes,thiscollectionof surveysshouldproveusefulnotonlyforseasonedresearchersinthefield,but alsoforbeginninggraduatestudentswhoareinterestedingettinguptospeed ontheworkinalgorithmicrandomnessthathasbeencarriedoutinthelast decadeaswellasfornonspecialistswhoarelookingforaccessiblepointsof entryintorecenttrendsinalgorithmicrandomness. Thesurveysinthisvolumeareorganizedasfollows. Webeginwith“Key vii viii PREFACE developmentsinalgorithmicrandomness”byJohannaFranklinandChristo- pherPorter,whichprovidesanaccountofthedevelopmentofthefield. This articleservestoprovideageneralintroductiontotheunderlyingconceptsof algorithmicrandomnessforthosenewtothesubject,emphasizingthework that serves as a background for the surveys in this volume via a historical perspective,andincludesathoroughintroductiontothenecessaryconcepts fromcomputabilitytheory. Inadditiontoadiscussionofthefundamental ideasofalgorithmicrandomnessfromthe1960sand1970s,weemphasizekey resultsinthe1980s,1990s,andearly2000s. Next, we have three articles on the interaction between algorithmic ran- domness and topics from classical analysis and ergodic theory. First, in “Algorithmicrandomnessinergodictheory,”HenryTowsnerlaysoutrecent workoninteractionsbetweenergodictheoryandvariousnotionsofalgorithmic randomness. Afterabriefreviewofthebasicsofdynamicalsystems,Towsner explainshowclassicalresultsonthetypicalpointsinergodicsystems,suchas variantsofBirkhoff’sergodictheorem,providecharacterizationsofdifferent classesofeffectivelytypicalpoints(often,thepointsthatareMartin-Lo¨fran- domorSchnorrrandom)whentheseresultsareframedintermsofcomputable transformations and particular types of effective open sets. Towsner also discussestheeffectivityoftheergodicdecompositionaswellasthecharacteri- zationoftypicalpointsinthenonergodicsetting. Inthelastseveralsections ofhispaper,Towsnerbrieflydiscussesnotionsofrandomnesswithrespectto nonuniformmeasures,notionsofrandomnessingeneralcomputablemetric spaces,andrandomnessinmoregeneraltopologicalsystems. Second, Jason Rute’s survey, “Algorithmic randomness and construc- tive/computablemeasuretheory,”providesadetailedhistoricalaccountofthe connectionsbetweenalgorithmicrandomnessandconstructivemathematics aswellasacatalogofresultsthatcharacterizevariousnotionsofalgorithmic randomnessintermsoftheoremsofclassicalanalysis. Forexample,astandard resultfromclassicalanalysisisthateverymonotonefunctionf :[0,1]→R isdifferentiableatalmosteveryx ∈ [0,1]. However,ifwerestrictourselves tocomputablefunctions,wefindthatx iscomputablyrandomifandonlyif everycomputable,monotonefunctionf :[0,1]→Risdifferentiableatx,a recentresultduetoBrattka,J.Miller,andNies. Aftersurveyingmanyresults of this flavor and discussing their connections to constructive and reverse mathematics,Ruteconcludeswithanextendeddiscussionoftheequivalence ofanumberofapproachestocomputablemeasuretheory. Third, Mathieu Hoyrup’s “Algorithmic randomness and layerwise com- putability”isanintroductiontothenotionoflayerwisecomputability,first introducedbyHoyrupandRojas. Motivatedasaformalizationofthenotionof aneffectivelymeasurablefunction,layerwisecomputablefunctionsaredefined explicitlyintermsofMartin-Lo¨frandomnessbymeansofauniversalMartin- Lo¨ftest. Hoyrupthendemonstratesthefruitfulnessofthisnotion,detailing PREFACE ix applicationstothestudyofalgorithmicrandomnesswithrespecttoaclass ofmeasures,thedecompositionofmeasures,theergodicdecomposition,and randomobjectssuchasalgorithmicallyrandomclosedsetsandalgorithmically randomBrownianmotion. The third grouping of surveys consists of more traditional topics in al- gorithmicrandomness,namely,relativerandomness,Omeganumbers,and randomnesswithrespecttononuniformprobabilitymeasuresonCantorspace. WhilethesetopicsaretreatedinbothDowneyandHirschfeldt’sandNies’s books, a number of new approaches to these topics that have changed the directionofthefieldhaveemergedsincethen. InJohannaFranklin’s“Rel- ativization in randomness,” she turns her attention to two different topics that exemplify recent work in this area: lowness for randomness and van Lambalgen’sTheorem. Afterasummaryofresultsobtainedusingthestandard relativization,sheconsidersrelativizationthroughtwodifferentlenses. The first is a study of the robustness of the class of sequences that are low for Martin-Lo¨frandomness(theK-trivials)bymeansofrecentcharacterizations usingavarietyofmethods. Hersecondfocusisthatofnewer,moreuniform approaches to relativization, namely those by Miyabe, Rute, and Kihara. Theseapproachesarestudiedviaaconsiderationoftheextenttowhichvan Lambalgen’sTheoremholdsfordifferenttypesofrandomsequencesunder thisnewrelativizationaswellasthepropertiesthatthesequencesthatarelow forthesetypesofrandomsequencesunderithave. Next, in “Aspects of Chaitin’s Omega,” George Barmpalias provides an accountoftherecentresearchonChaitin’scelebratednumberΩ. Inaddition tosurveyinganumberofstandardresultsonΩ,includingmanyresultsthat wereestablishedintheearly-tomid-2000saswellassomeofChaitin’soriginal work,Barmpaliasalsocoversawiderangeofnewerresults,includinganalogs of Chaitin’s Omega in the computably enumerable sets, various results on theratesofapproximationsofOmeganumbers,thecomputationalpowerof Omeganumbers,andprobabilitiesassociatedwithdifferentclassesofmachines andtheirrelationshipstorelativizationsofOmega. In“Biasedalgorithmicrandomness,”ChristopherPorterroundsoutthissec- tionwithasurveyofrecentworkinvolvingalgorithmicrandomnesswithrespect tononuniformprobabilitymeasuresonCantorspace,atopicthatfirstappeared inMartin-Lo¨f’spivotal1966paperandwasfurtheredbyLevinandZvonkin andthenbyKautzinhisdissertation. Thesurveydealswithtwomaintopics: (1)randomnesswithrespecttocomputableprobabilitymeasures,including transformingrandomnessfromonemeasureintorandomnessinanother,the computationalpowerofsequencesthatarerandomwithrespecttoacomputable measure,andrandomnesswithrespecttocomputable,countablysupported measures,and(2)randomnesswithrespecttononcomputablemeasures,in- cludingtheworkofReimannandSlamanoncontinuousrandomness,neutral measures,andthenotionofblindrandomnessdevelopedbyKjos-Hanssen.
The list of books you might like
