Bridging Constraint Satisfaction and Boolean Satisfiability

121 Pages·2015·1.661 MB·English

by  Justyna Petke (auth.)

Artificial Intelligence: Foundations, Theory, and Algorithms
Justyna Petke
Bridging Constraint
Satisfaction and
Boolean Satisfiability
Artificial Intelligence: Foundations, Theory,
and Algorithms
Serieseditors
BarryO’Sullivan
Cork,Ireland
MichaelWooldridge
Oxford,UnitedKingdom
Moreinformationaboutthisseriesathttp://www.springer.com/series/13900
Justyna Petke
Bridging Constraint
Satisfaction and Boolean
Satisfiability
123
JustynaPetke
Dept.ofComputerScience
UniversityCollegeLondon
London,UnitedKingdom
ArtificialIntelligence:Foundations,Theory,andAlgorithms
ISBN978-3-319-21809-0 ISBN978-3-319-21810-6 (eBook)
DOI10.1007/978-3-319-21810-6
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Preface
A wide range of problems can be formalized as a set of constraints that need to
besatisfied.Infact,suchamodeliscalledaconstraintsatisfactionproblem(CSP).
Another way to represent a problem is to express it as a formula in propositional
logic,or,inotherwords,aBooleansatisfiabilityproblem(SAT).Inthequesttofind
efficientalgorithmsforsolvinginstancesofCSPandSATspecialisedsoftwarehas
beendeveloped.Itis,however,notclearwhenshouldwechooseaSAT-solverovera
constraintsolver(andviceversa).CSP-solversareknownfortheirdomain-specific
reasoning, whereas SAT-solvers are considered to be remarkably fast on Boolean
instances. In this book we tackle these issues by investigating the connections
betweenCSPandSAT.
In order to answer the question why SAT-solvers are so efficient on certain
classes of CSP instances, we first present the various ways one can encode a
CSP instance into SAT. Next, we show that with some encodings SAT-solvers
simulate theeffectsofenforcinga formoflocalconsistency,calledk-consistency,
inexpectedpolynomial-time.ThusSAT-solversareabletosolveCSPinstancesof
bounded-width structure efficiently in contrast to conventional constraint solvers.
By consideringthe variousways one can encodeCSP domainsinto SAT, we give
theoretical reasons for choosing a particular SAT encoding for several important
classes of CSP instances. In particular, we show that with this encoding many
probleminstancesthatcanbesolvedinpolynomial-timewillstillbeeasilysolvable
once they are translated into SAT. Furthermore, we show that this is not true for
severalotherencodings.
Finally, we compare the various ways one can use a SAT-solver to solve the
classical problem of the pigeonhole principle. We perform both theoretical and
empiricalcomparisonofthevariousencodings.Weconcludethatnoneoftheknown
encodingsfortheclassicalrepresentationoftheproblemwillresultinanefficiently
solvableSATinstance.Thusinthiscaseconstraintsolversareamuchbetterchoice.
v

Acknowledgements
ThisbookarosefrommyDPhilthesis‘OntheBridgeBetweenConstraintSatisfac-
tionandBooleanSatisfiability’awardedbytheUniversityofOxfordin2012.
First and foremost I would like to thank my supervisor Peter Jeavons for his
enthusiasticencouragementandvaluableguidancethroughoutmydoctoratestudies.
Icouldnothaveimaginedhavingabettersupervisor.
I thank my colleagues from the Oxford Constraints group: Stanislav Živný,
András Salamon, Christopher Jefferson, Karen Petrie, Páidí Creed, Evgenij
Thorstensen,ConradDrescher,MarkusAschingeraswellasDavidCohen.Iwould
alsoliketothankmycollegeadvisorJoëlOuaknine.
I am especially thankful to Stanislav Živný with whom I shared an office
throughoutmydoctoralstudies.Iamgratefulforhisadviceandencouragement.
I wouldalso like to thankmyexaminers,GeorgGottlob andBarry O’Sullivan,
forprovidinghelpfulcorrectionsaswellasmyformersupervisorRobinHirschand
DanielHulmeforgettingmeintouchwiththeOxfordConstraintsgroupinthefirst
place.
The provision of a Doctoral Training Award by the UK Engineering and
Physical Sciences Research Council is greatly appreciated. I am also grateful to
the Departmentof ComputerScience and St John’s College, Oxford,for financial
support.
Finally,Iwouldliketothankmyfamilyandfriends.Myparentsandgrandparents
havebeena constantsourceofsupportandthisbookandthe relatedthesis would
certainlynothaveexistedwithoutthem.
vii

Contents
1 Introduction .................................................................. 1
2 Background................................................................... 5
2.1 Constraintsatisfactionproblem(CSP)................................. 6
2.2 CSP-solvers ............................................................. 8
2.2.1 Search ........................................................... 9
2.2.2 Constraintpropagation ......................................... 11
2.2.3 Combiningpropagationandsearch............................ 15
2.3 Booleansatisfiabilityproblem(SAT) .................................. 15
2.4 SAT-solvers.............................................................. 15
2.4.1 Search ........................................................... 17
2.4.2 Booleanconstraintpropagation(BCP) ........................ 18
2.4.3 Conflictanalysis ................................................ 19
2.5 HybridsolversandSMT-solvers ....................................... 21
2.6 Summary ................................................................ 21
3 SolverperformanceontractableCSPs:empiricalevaluation .......... 25
3.1 Preliminaries ............................................................ 27
3.2 Constant-closedconstraints............................................. 28
3.3 Max-closedconstraints ................................................. 31
3.4 0/1/allconstraints ....................................................... 32
3.5 Connected-row-convexconstraints..................................... 35
3.6 Bounded-widthstructures .............................................. 37
3.7 Summary ................................................................ 40
4 SATencodings................................................................ 43
4.1 Sparseencodings........................................................ 43
4.2 Thelogencoding........................................................ 46
4.3 Regularandinterval-basedencodings ................................. 47
4.4 Otherencodings......................................................... 49
4.5 WhatisagoodSATencoding?......................................... 51
4.6 Summary ................................................................ 53
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