Combined Decision Procedures for Nonlinear Arithmetics, Real and Complex

318 Pages·2012·2.25 MB·English

Combined Decision Procedures for Nonlinear
Arithmetics, Real and Complex
Grant Olney Passmore
U N I VE
R
E S
I
H
T
T Y
O H
G
F
R
E U
DI N B
Doctor of Philosophy
Mathematical Reasoning Group
Algorithms and Complexity Group
LFCS, School of Informatics
University of Edinburgh
2011

Abstract
We describe contributions to algorithmic proof techniques for deciding the satisfia-
bility of boolean combinations of many-variable nonlinear polynomial equations and
inequalitiesovertherealandcomplexnumbers.
In the first half, we present an abstract theory of Gro¨bner basis construction al-
gorithms for algebraically closed fields of characteristic zero and use it to introduce
and prove the correctness of Gro¨bner basis methods tailored to the needs of modern
satisfiability modulo theories (SMT) solvers. In the process, we use the technique of
proof orders to derive a generalisation of S-polynomial superfluousness in terms of
transfinite induction along an ordinal parameterised by a monomial order. We use this
generalisationtoprovetheabstract(“strategy-independent”)admissibilityofanumber
ofsuperfluousS-polynomialcriteriaimportantforefficientbasisconstruction. Finally,
weconsiderlocalnotionsofproofminimalityforweakNullstellensatzproofsandgive
ideal-theoreticmethodsforcomputingcomplex“unsatisfiablecores”whichcontribute
toefficientSMTsolvinginthecontextofnonlinearcomplexarithmetic.
In the second half, we consider the problem of effectively combining a heteroge-
neous collection of decision techniques for fragments of the existential theory of real
closed fields. We propose and investigate a number of novel combined decision meth-
odsandimplementtheminourprooftoolRAHD(RealAlgebrainHighDimensions).
We build a hierarchy of increasingly powerful combined decision methods, culminat-
inginageneralisationofpartialcylindricalalgebraicdecomposition(CAD)whichwe
call Abstract Partial CAD. This generalisation incorporates the use of arbitrary sound
butpossiblyincompleteproofproceduresfortheexistentialtheoryofrealclosedfields
as first-class functional parameters for “short-circuiting” expensive computations dur-
ing the lifting phase of CAD. Identifying these proof procedure parameters formally
with RAHD proof strategies, we implement the method in RAHD for the case of
full-dimensional cell decompositions and investigate its efficacy with respect to the
Brown-McCallumprojectionoperator.
Weendwithsomewishesforthefuture.
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Acknowledgements
Pursuing this work has been for me some kind of paradise. There are so many
whomIwishtothank.
Firstandforemost,IthankmyPhDsupervisor,PaulB.Jackson. Paul’sencourage-
ment,direction,intellectualdexterityandendlesspositivityhavemadethisdissertation
ajoytocompose. IfindithardtoimaginehowonecouldhaveabetterPhDsupervisor
than mine. Paul was ever accessible, always found time to assist me with unexpected
difficulties(technicalorotherwise),anddidmuchtohelpmefeelathomeinaforeign
landbyincludingmeinsomeofhisandElizabeth’sspecialfamilyactivities. Nomatter
how discouraged I might have been by a particular aspect of my work, I could always
count on Paul to find the hidden morsels of progress worth celebrating and pursuing.
Fromthisexperience,IhaveatemplateforthekindofsupervisorIhopeIwillbewhen
I have PhD students. In addition to our rich personal collaboration, Paul also made it
possibleformetotakevisitingpositionsatSRIInternational,MicrosoftResearchand
INRIA. Without his support in this, much of my thesis work would not have come to
fruition. Iamverypleasedwehavereceivedafour-yearEPSRCgranttocontinuethis
worksothatourcollaborationshallgoonformanyyearstocome.
I thank Leonardo de Moura of Microsoft Research, Washington. Leo’s friendship
has had an immeasurable impact on me and on the work contained in my thesis. It
was Leo’s idea to focus on developing Gro¨bner basis methods tailored to the needs of
industrial-strength SMT solving, and this goal of his (which has since become also a
goal of mine) led us down beautiful paths, many of which are still unwinding. The
countless days and nights we spent developing these techniques, making and revising
conjectures,andatlastprovingourlongsoughtaftertheoremsremainwithmeassome
of my favourite memories of my life. I find the sheer intensity of our collaboration
impossible to describe. As we are in the throws of writing two more papers together
asItypethis,IamhappythatmylongcollaborationwithLeoisonlyjustbeginning.
IthankNShankarandSamOwreofSRIInternational. Theyareincredibleteachers
who have given me so much. The first idea for my RAHD system for making exis-
tential decisions over real closed fields was developed while I was a Visiting Fellow
at SRI under Shankar and Sam during May - October, 2008. During this visit, I wrote
theinitialRAHDprototypeasanextensionoftheproofassistantPVS.Shankartaught
me much about research and how to place it in the context of an artful and fulfilling
life. Sam taught me a tremendous amount about almost everything — his lessons on
Lisp, jazz, racquetball, go and snooker remain especially vibrant in my mind. There
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aremanyafternoonsIwishIcouldwalkintoSam’sofficetoseekhisLispadviceover
agourdofyerbamate´ whileunderthespellofablastingCecilTaylorrecord. Paradise.
At SRI, I was inspired by numerous helpful discussions with John Rushby, Bruno
DutertreandAshishTiwari. Ithankthemforthis. IalsothanktheUSNationalScience
FoundationandNASAforfundingmySRIfellowship. Isendaspecialthankyoutothe
Berkeley SMASH summer mathematics camp for giving me the chance to share with
thoseunfathomablybrighthighschoolstudentsthebeautyofmathematics(andofreal
algebraic decision problems, in particular!). At SRI, my friendships with Max Meier
and Florent Kirchner were especially edifying. Further, I am grateful to Florent and
INRIA/IRISA for funding my month-long position as a Visiting Researcher at INRIA
in Rennes, Bretagne, France in April, 2010, where Florent and I began our work of
connectingRAHDandtheproofassistantCoq. AlsoonthetopicoftheFrench,Ihave
benefited very much from conversations with Yves Bertot and Assia Mahboubi and I
thankthemfortheiradviceandencouragement.
I thank my fellow LFCS PhD students, Julian Guiterrez, Willem Heijltjes, Ohad
Kammar, Gavin Keighren and Matteo Mio. Our regular lunches, pints, musical hangs
and invigorating conversations have done much to keep me going. I am thankful to
KoushaEtessami andLeonidLibkinfor theirtremendoushelp, especiallyintheirrole
onmyyearlyprogressreviewcommittee. IalsothankJeffEgger,AlexSimpson,John
Longley, Lorenzo Clemente, Ben Kavanagh, Sarah Luger, Annette Leonhard, Gaya
NadarajanandTeresaLlanofortheirfriendshipandadvice. Iamespeciallythankfulto
GianmariaSilvello,whosenine-monthvisittoEdinburgh—livedequallybetweenour
shared office, The Jazz Bar, Dario’s and the Film House Cinema — had a tremendous
positiveimpactonme. ToGianmaria,Isayoneword: Legendary.
I thank The Edinburgh Mathematical Reasoning Group for allowing me to under-
take my thesis work in such a welcoming and dedicated community. I’ve benefited
fromnumerousdiscussionswithAlan Bundy,LucasDixon,JacquesFleuriot,Andrew
Ireland, Alan Smaill, Ewen MacLean and Phil Scott over the years and I thank them
for this. When my PhD student funding ran out and our grant application was still
under review, Alan Bundy found a way to fund me on the DReaM group’s Platform
Grant — without his help, I would have been in serious trouble. I am also grateful to
theScottishTheoremProversSeminarandlookforwardtomycontinuedinvolvement.
Before I began in Edinburgh, I spent the year 2006-2007 at the Mathematical Re-
search Institute in The Netherlands under Jaap van Oosten and Ieke Moerdijk. This
year-long Master Class in Mathematical Logic was vital for my mathematical devel-
v
opmentandIthankmyteachersatMRI,JaapvanOosten,IekeMoerdidk,HenkBaren-
dregt, Herman Guevers, Bas Spitters, Bas Terwijn, Wim Veldman and Albert Visser.
I am especially grateful to my friends and fellow students at MRI, David Carchedi,
Yves Fomatati, Danko Iliik, Johanny Suarez, Takako Nemoto and Andrew Polonsky.
While in Holland, I taught a mechanical theorem proving course with Joost J. Joosten
attheInstituteforLogic,LanguageandComputationofUniversityofAmsterdam,and
IthankJoostandourremarkablestudents. Myclosefriendshipandcollaborationwith
JoosthasdonemuchtofuelmethroughoutmyPhD.ToJoost,Isay: TMNFS2K7AB.
As an undergraduate at the University of Texas at Austin, I have Bob Boyer, Josh
Dever, Matt Kaufmann, Greg Lavender, Vladimir Lifschitz, J Moore and Altha Rodin
mosttothankforguidingandencouragingmymathematicalinterests. Theyhavemade
a tremendous impact on my life. I am especially grateful for my deep friendship with
Denis Ignatovich; he continues to be a pivotal source of wisdom. I thank Jeremy Avi-
gad for having me in his 2005 NSF Summer School in Proof Theory at the University
ofNotreDame. ThiscoursewasthefirsttimeIlearnedofthedecidabilityofthetheory
of real closed fields; I was mesmerised by it then and I am mesmerised by it now. I
also thank Tom Ball, Daniel Brown, Yuri Gurevich, John Harrison, Rustan Leino and
Andra´sSalamonfortheirfriendshipandmanyintellectualgifts.
While I was a PhD student, I wrote two albums with my close friend Barry De-
Bakey which we recorded in Austin, Texas with the incredible help of Dave and Eddy
Hobizal. The first album, “Olney Clark,” was produced by Eddy and was especially
important for me as a musical outlet to document the process of doing my PhD. Our
friendshipsandmusicalcollaborationscontinuetoprovidemewithcrucialinspiration.
The final bits of my thesis were completed at Cambridge University after I began
my RA position on our joint Cambridge-Edinburgh EPSRC grant “Automatic Proof
Procedures for Polynomials and Special Functions.” I am very grateful to Larry Paul-
son for his advice and encouragement. I thank the Cambridge Automated Reasoning
Group as well as Thomas Forster and the Cambridge Set Theory Group for providing
such an open and invigorating environment. At Cambridge, I also thank Mike Gordon
and my office-mates, James Bridge, Will Denman and Magnus Myreen. Furthermore,
I thank my PhD examiners, Daniel Kroening at Oxford and Alan Smaill at Edinburgh
fortheircarefulreading,livelydiscussionandmosthelpfulsuggestions.
Finally,Ithankmyfamily. TomyparentsDonnaandJohn,mysistersStarr,Jacque
and Skye: You have loved me into being, you have made me who I am. To you and to
Erika: Ithankyouforyourlove,yourkindnessandyourunwaveringbeliefinme.
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Declaration
I declare that this thesis was composed by myself, that the work contained herein is
myownexceptwhereexplicitlystatedotherwiseinthetext,andthatthisworkhasnot
beensubmittedforanyotherdegreeorprofessionalqualificationexceptasspecified.
Ihavebenefitedgreatlyfromcollaborationwithmyco-authorsPaulB.Jacksonand
Leonardo de Moura. Much work contained herein is a product of such collaborations.
At the end of the introductory section of each chapter containing joint work, I provide
an explanation of the contributions each co-author made to the work presented in that
chapter. When appropriate, I also give references to our published papers in which
suchworkappears.
(GrantOlneyPassmore)

ToDonnaandJohn,Starr,JacqueandSkye.
Since a decision method, by its
very nature, requires no intel-
ligence for its application, it is
clear that, whenever one can give
a decision method for a class K
of sentences, one can also devise
a machine to decide whether an
arbitrarysentencebelongstoK.
It often happens in mathematical
research, both pure and applied,
that problems arise as to the
truth of complicated sentences of
elementary algebra or geometry.
The decision method presented in
this work gives the mathematician
the assurance that he will be able
to solve every such problem by
workingatitlongenough.
– Alfred Tarski, “A Decision
Method for Elementary Algebra
andGeometry,”1948.