Small Dynamic Complexity Classes: An Investigation into Dynamic Descriptive Complexity

156 Pages·2017·2.678 MB·English

by  Thomas Zeume (auth.)

Thomas Zeume
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An Investigation into Dynamic Descriptive Complexity
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Lecture Notes in Computer Science 10110
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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
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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