Efficient Solving of Large Arithmetic Constraint Systems with Complex Boolean Structure: Proof Engines for the Analysis of Hybrid Discrete-Continuous Systems

182 Pages·2011·0.91 MB·English

by  Christian Herde

Christian Herde
Efficient Solving of Large Arithmetic Constraint Systems
with Complex Boolean Structure
VIEWEG+TEUBNER RESEARCH
Christian Herde
Efficient Solving
of Large Arithmetic Constraint
Systems with Complex Boolean
Structure
Proof Engines for the Analysis
of Hybrid Discrete-Continuous Systems
With a foreword by Prof. Dr. Martin Fränzle
VIEWEG+TEUBNER RESEARCH
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Dissertation Universität Oldenburg, 2010
1st Edition 2011
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Foreword
Over the past decades, we have seen a slow, but steady popularization of automatic
verification technology — once the Holy Grail of formal methods in computer sci-
ence—thatisindicativeofthematuritythefieldhasachieved. Onlyfewuniversities
do still hesitate to expose undergraduates to the field’s basic methods, the pertinent
tools, and their underlying algorithms, and industrial takeover is undeniably gain-
ingimpetus,followingeducationwiththenaturalphasedelay. Overtheyears,some
basictechniqueshavebecomecornerstonesofthefield,formingidentifiableandreap-
pearing building blocks of the plethora of tools that have been built. Among these
fundamentals are abstract interpretation, binary decision diagrams, and satisfiability
solving, tonamejustafew. Withinthisbook, youwillfindaninformedaccountof
a number of substantial contributions to the latter field, in particular addressing the
domain of satisfiability modulo theories, which has become instrumental to various
mechanic verification schemes in hardware and software validation. Despite being
basedonaPhDthesis,whichIhadthepleasuretoadvise,thebookelaboratesinequal
detailontheunderlying“folklore”ideasandtechniquesandontheauthor’sowncon-
tributions,complementingbothbyextensivepointerstoopenproblemsandideasfor
further research. I hope you as a reader will find it helpful, no matter whether you
are a novice trying to understand satisfiability solving for complex-structured arith-
meticconstraintsorareanexpertlookingforaclear-cutdelineationofthetechniques
developed in the Transregional Collaborative Research Center AVACS (Automatic
VerificationandAnalysisofComplexSystems,fundedbytheDeutscheForschungs-
gemeinschaft)fromcompetingapproaches.
MartinFra¨nzle

Acknowledgements
In a sense, nature has been continually computing the ‘next state’ of the
universe for billions of years; all we have to do – and, actually, all wecan
do–is‘hitcharide’onthishugeongoingcomputation,andtrytodiscover
whichpartsofithappentogoneartowherewewant.
TommasoToffoli,1982
This thesis marks a major waypoint on my excursion into the field of formal verifi-
cationandconstraintsolving,apleasurableridealongabranchofthe‘hugeongoing
computation’whichturnedouttobeoneofthemoreexcitingones. Iwouldliketo
thankallthosewhomadethisjourneypossible:
First and foremost, I would like to thank my supervisor Martin Fra¨nzle. A big
thankyou,Martin,forallIhavelearnedfromyou,foryourperpetualwillingnessto
discussanyquestionsandideasIhavehad, foryourvaluableadvice, andforgiving
methetimeIneededtofinishthiswork.
ParticularthanksgotoBerndBeckerforkindlyacceptingtoco-examinethisthe-
sis,andtohimandthemembersofhisresearchgroupfortheirhospitalityandsupport
duringmyvisitstoFreiburg. Ialsoextendmythankstomyothercommitteemem-
bersSibylleFro¨schleandAndreasWinterfortakingtimetoparticipateinmydefense.
Moreover,IwouldliketoexpressmygratitudetoWernerDammforsparkingmyin-
terestinsafety-criticalsystemsandforofferingmemyfirstjobinformalverification
duringanoralexammanyyearsago.
IamparticularlyindebtedtomyfriendsandcolleaguesfromtheUniversityofOl-
denburg,fromtheTransregionalCollaborativeResearchCenterAVACS,andfromthe
OFFIS Institute for Information Technology for providing an inspiring and pleasant
workingenvironment. IamespeciallygratefultoAndreasEggersandTinoTeigefor
outstanding teamwork, for many discussions on research and life, and for the great
timewespenttogether.
Lastly,andmostimportantly,IamdeeplythankfultomygrandparentsAnnaand
Friedrich Bo¨deker who raised me and gave me loving care and support in difficult
times.Totheirmemory,Idedicatethisthesis.
ChristianHerde

Abstract
Duetotheincreasinguseofmoreandmorecomplexcomputerizedsystemsinsafety-
critical applications, formal verification of such systems is of growing importance.
Among the most successful methods in formal verification of finite-state systems is
bounded model checking (BMC), a technique for checking whether an unsafe sys-
tem state is reachable within a fixed number of steps. BMC belongs to a class of
verificationalgorithmshavingincommonthattheverificationtaskisreducedtothe
problem of checking the satisfiability of a propositional formula or a series thereof.
Thoughoriginallyformulatedfordiscretetransitionsystemsonly,BMCisinprinci-
ple also applicable to hybrid discrete-continuous systems, which naturally arise e.g.
inthefieldofembeddedsystemswheredigital(discrete)controllersarecoupledwith
analog (continuous) physical plants. The BMC formulae arising from such systems
are,however,nolongerpurelypropositional,butusuallycomprisecomplexBoolean
combinations of arithmetic constraints over real-valued variables, thus entailing the
needfornewdecisionprocedurestosolvethem.
Thisthesisdealswiththedevelopmentofsuchprocedures. Akeycomponentof
thealgorithmswepresentistheDPLLprocedureforsolvingBooleanformulaewhich
areinconjunctivenormalform(CNF).Asafirstcontribution,wedemonstratethatthe
accelerationtechniques,whichenabledtheenormousperformancegainsofBoolean
SAT solvers in the recent past, generalize smoothly to DPLL-based procedures for
solvingsystemsofpseudo-Booleanconstraints,amuchmoreconciserepresentation
of Boolean functions than CNFs. Second, we investigate how to efficiently couple
a linear programming routine with a DPLL-based SAT solver in order to obtain a
solverwhichistailoredforBMCofhybridsystemswithlineardynamics. Inpartic-
ular,weexaminehowtoexploittheuniquecharacteristicsofBMCformulaeandthe
incremental nature of BMC for various optimizations inside the solver. Finally, we
presentourmaincontribution, atightintegrationoftheDPLLprocedurewithinter-
valconstraintsolving, resultinginanalgorithm, called ISAT,whichgeneralizesthe
DPLLprocedureandiscapableofsolvingarbitraryBooleancombinationsofnonlin-