Small Dynamic Complexity Classes: An Investigation into Dynamic Descriptive Complexity PDF
Preview Small Dynamic Complexity Classes: An Investigation into Dynamic Descriptive Complexity
Thomas Zeume 0 Small Dynamic 1 1 0 1 S Complexity Classes C N L An Investigation into Dynamic Descriptive Complexity 123 Lecture Notes in Computer Science 10110 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison, UK Takeo Kanade, USA Josef Kittler, UK Jon M. Kleinberg, USA Friedemann Mattern, Switzerland John C. Mitchell, USA Moni Naor, Israel C. Pandu Rangan, India Bernhard Steffen, Germany Demetri Terzopoulos, USA Doug Tygar, USA Gerhard Weikum, Germany FoLLI Publications on Logic, Language and Information Subline of Lectures Notes in Computer Science Subline Editors-in-Chief Valentin Goranko, Stockholm University, Sweden Michael Moortgat, Utrecht University, The Netherlands Subline Area Editors Nick Bezhanishvili, University of Amsterdam, The Netherlands Anuj Dawar, University of Cambridge, UK Philippe de Groote, Inria Nancy, France Gerhard Jäger, University of Tübingen, Germany Fenrong Liu, Tsinghua University, Beijing, China Eric Pacuit, University of Maryland, USA Ruy de Queiroz, Universidade Federal de Pernambuco, Brazil Ram Ramanujam, Institute of Mathematical Sciences, Chennai, India More information about this series at http://www.springer.com/series/7407 Thomas Zeume Small Dynamic Complexity Classes An Investigation into Dynamic Descriptive Complexity 123 Author ThomasZeume Fakultät für Informatik TU Dortmund Dortmund, Nordrhein-Westfalen Germany Thiswork wascarried out at: Department ofComputer Science, TU DortmundUniversity inDortmund, Germany andaccepted there asaPhD thesis. ISSN 0302-9743 ISSN 1611-3349 (electronic) Lecture Notesin Computer Science ISBN 978-3-662-54313-9 ISBN978-3-662-54314-6 (eBook) DOI 10.1007/978-3-662-54314-6 LibraryofCongressControlNumber:2017930617 LNCSSublibrary:SL1–TheoreticalComputerScienceandGeneralIssues ©Springer-VerlagGmbHGermany2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynow knownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbookare believedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsortheeditors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictionalclaimsin publishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringer-VerlagGmbHGermany Theregisteredcompanyaddressis:HeidelbergerPlatz3,14197Berlin,Germany Preface On the first few days of my PhD studies in the summer of 2009, my adviser Thomas Schwentick introduced me to dynamic complexity. Very recently Wouter Gelade, Marcel Marquardt, and Thomas had obtained a very nice characterization of regular languages in terms of dynamic complexity, and also a lower bound for the dynamic complexity of the alternating reachability problem. Many interesting problems in this area seemed to be awaiting a solution; and so I attempted to prove a lower bound for reachability. I was not successful. After two (at the end frustrating) months, I aban- doned this project. In the following two-and-a-half years I almost forgot about dynamic complexity. Decidability issues for thetwo-variable fragmentoffirst-order logicturnedouttobea muchmoreaccessibleandfruitfulfield.ThomasandIobtainedpromisingresults,anda PhDinthisfielddidnotseemtobetoofaraway.ThiswasthemomentwhenThomas askedwhetherIwouldbeinterestedinapplyingforfundsfromtheDFG.Ifsuccessful, such funding could relieve me from my teaching obligations. Wedecidedtohaveasecond,deeperlookintodynamiccomplexity,andtoapplyfor funds for an extensive study of the power of logics in dynamic settings. At that time, thedecisiontospendmoretimeondynamiccomplexitywasnoteasyforme.Iwasin the third year of my PhD and already had results and further ideas for two-variable logics;anditwasnotclearwhetheranapplicationforfundingwouldbesuccessful.On the other hand, I now had more experience, which might be helpful for attacking the verysameproblemsthatIhadtriedtosolveatthebeginningofmyPhD.Idonotregret the decision. WiththethesisathandIwanttodocumenttheprogressindynamiccomplexitythat we have made in the last two-and-a-half years. The focus of this thesis is on small dynamic descriptive complexity classes, in particular on lower bound methods for them. A short summary of results on decidability issues for two-variable logic is presented at the end of the thesis. IamverygratefultoThomasSchwentickforallhissupportthroughouttheyearsofmy PhDstudies,andforbeingagreatexampleofhowtobearesearcherandteacher.Ithankthe refereesErichGrädelandThomasSchwentickaswellasCorneliaTadrosandJensTeubner fortheirworkinmydefensecommittee.Further,IthankSamirDatta,SebastianSiebertz, andNilsVortmeierformanyfruitfuldiscussionsaboutdynamiccomplexity.Ialsothank thenumerouscolleaguesatDortmundandinthelogicanddatabasecommunityformaking VI Preface the last years a great time. Moreover, I thank Katja Losemann and Nils Vortmeier for proofreadingpartsofthiswork.IacknowledgethefinancialsupportbytheGermanDFG undergrantSCHW678/6-1. My warmest thanks goes to my family, to Katja, and to all my friends who sup- ported me during the past couple of years. January 2015 Thomas Zeume Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Dynamic Complexity: Definitions and Examples. . . . . . . . . . . . . . . . 11 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The Dynamic Complexity Framework. . . . . . . . . . . . . . . . . . . . . . 13 2.3 Three Basic Dynamic Complexity Classes . . . . . . . . . . . . . . . . . . . 15 2.3.1 The Class DYNFO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.2 The Class DYNPROP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.3 The Class DynQF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Variants of the Dynamic Complexity Framework . . . . . . . . . . . . . . 21 2.5 A Case Study: Graph Queries. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5.1 Regular Path Queries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5.2 Beyond Regular Path Queries. . . . . . . . . . . . . . . . . . . . . . . 27 2.6 Outlook and Bibliographic Remarks . . . . . . . . . . . . . . . . . . . . . . . 34 3 Relating Small Dynamic Complexity Classes. . . . . . . . . . . . . . . . . . . 35 3.1 A Hierarchy of Dynamic Classes . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.1.1 Tools for Collapsing Dynamic Classes. . . . . . . . . . . . . . . . . 42 3.1.2 Eliminating Negations and Inverting Quantifiers . . . . . . . . . . 45 3.1.3 Eliminating Disjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.4 Simulating Functions by Conjunctive Queries. . . . . . . . . . . . 57 3.2 Short Interlude: D-Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 Relating Dynamic Classes and Static Classes . . . . . . . . . . . . . . . . . 67 3.3.1 A Dynamic Characterization of First-Order Logic. . . . . . . . . 69 3.3.2 DYNPROP Captures Semi-positive 9(cid:2)FO Under Insertions . . . . 73 3.4 Eliminating Built-In Arithmetic. . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.5 Outlook and Bibliographic Remarks . . . . . . . . . . . . . . . . . . . . . . . 80 4 Lower Bounds for Dynamic Complexity Classes. . . . . . . . . . . . . . . . 83 4.1 Quantifier-Free Update Programs . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.1.1 The Substructure Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.1.2 Applications of the Substructure Lemma . . . . . . . . . . . . . . . 91 4.1.3 An Arity Hierarchy for Quantifier-Free Programs . . . . . . . . . 106 4.1.4 Fragments of Quantifier-Free Programs . . . . . . . . . . . . . . . . 108 VIII Contents 4.2 Quantifier-Free Update Programs with Functions . . . . . . . . . . . . . . 112 4.2.1 A Generalization of the Substructure Lemma . . . . . . . . . . . . 114 4.2.2 Applying the Generalized Substructure Lemma. . . . . . . . . . . 116 4.2.3 Why Lower Bounds for Binary Functions Are Hard to Prove . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3 First-Order Update Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.3.1 Applying Static Lower Bound Methods. . . . . . . . . . . . . . . . 125 4.3.2 Two Approaches for Restricted Initializations. . . . . . . . . . . . 131 4.4 Outlook and Bibliographic Remarks . . . . . . . . . . . . . . . . . . . . . . . 136 5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Chapter 1 Introduction 1.1 Introduction Inmanyoftoday’sdatamanagementscenariosthedataissubjecttofrequentmod- ifications, and it is often essential to react to those changes quickly. When a train iscanceledonshortnotice,travelersneedtofindalternativeconnectionsasfastas possible.Whenawebserveristemporarilynotavailable,datapackageshavetobe rerouted immediately. Also data in social networks is subject to frequent changes: modificationsoftherelationshipsofusersleadtonumerousconsequencesincluding thenecessityofupdatingthevisibilityofsensitivedata. Recomputationofaqueryresultfromscratchaftereachsmallchangeofthedata isoftennotpossibleinsuchscenariosduetothelargeamountofdataathandand efficiency considerations. Very often it is also not necessary: the breakdown of a singletraindoesaffectonlyaverysmallfractionofthewholetrainnetwork.Thus itisreasonabletotrytodynamicallyupdateessentialinformationinanincremental fashionbyreusinginformationthathasbeencomputedalreadybefore.Ideallysuch adynamicupdateshoulduselessresourcesthanrecomputationfromscratch. Approaches for such dynamic updates often store, besides the relevant data, additional information in order to facilitate the update process. This information is called auxiliary data. When updating the result of a query after a modification ofthedataoccurred,anupdateprocesshasaccesstoboththemodificationandthe stored auxiliary data. The auxiliary data, however, has to be updated as well. In Fig.1.1thedynamicpointofviewisjuxtaposedtotheclassicalstaticpointofview. Twofundamentallydifferentapproachesfordynamicallyupdatingtheresultofa queryhavebeenstudied,analgorithmicapproachandadeclarativeapproach. The algorithmic approach is not subject of this work. In this approach the goal istodevelopalgorithmsthatneedlessresourcesforrecomputingqueryresultsafter modificationsthananaïvealgorithmthatrecomputesresultsfromscratch.Agood startingpointforreadersinterestedinupperboundsfordynamicalgorithmsis[RZ08, DI08];agoodstartingpointforlowerboundtechniquesisthesurveybyMiltersen oncellprobecomplexity[Mil99]. ©Springer-VerlagGmbHGermany2017 1 T.Zeume,SmallDynamicComplexityClasses,LNCS10110 DOI:10.1007/978-3-662-54314-6_1
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