Extending Superposition with Integer Arithmetic, Structural Induction, and Beyond

136 Pages·2017·2.29 MB·English

by  Simon Cruanes

Extending Superposition with Integer Arithmetic,
Structural Induction, and Beyond
Simon Cruanes
To cite this version:
Simon Cruanes. Extending Superposition with Integer Arithmetic, Structural Induction, and Beyond.
Logic in Computer Science [cs.LO]. École polytechnique, 2015. English. ￿NNT: ￿. ￿tel-01223502￿
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ÉcoleDoctoraledel’Écolepolytechnique
INRIA
THÈSE DE DOCTORAT
Présentéepar
SimonCruanes
Pourobtenirlegradede
DOCTEURdel’ÉCOLEPOLYTECHNIQUE
Spécialité:Informatique
Soutenuele10Septembre2015
Extending Superposition with Integer
Arithmetic, Structural Induction, and Beyond
Directeursdethèse:
M.GillesDOWEK DirecteurdeRecherche,Inria
M.GuillaumeBUREL MaîtredeConférence,ENSIIE
Rapporteurs:
M.StephanSCHULZ Professeur,DHBWStuttgart
M.UweWALDMANN DirecteurdeRecherche,MPII,Saarbrücken
Examinateurs:
M.NicolasPELTIER ChargédeRecherche,CNRS
M.StéphaneGRAHAM-LENGRAND ChargédeRecherche,CNRS
M.SylvainCONCHON Professeur,UniversitéParis-Sud
Thisworkislicensedunderthecreativecommonsby-ndlicense.
Contents
1 Introduction 2
2 TechnicalPreliminaries 5
2.1 MathematicalConcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 BooleanLogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 First-Orderlogic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.3 Formulas,Literals,andClauses . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.4 Semantics:theCentralNotionofModel . . . . . . . . . . . . . . . . . . . . 15
2.4 Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 InferenceandSimplificationRules . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2 TheCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.3 RedundancyCriteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 AVATAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 ImplementingSuperpositioninaModularWayandExtendingIt 26
3.1 Logtk:AModularLibraryforFirst-OrderLogic . . . . . . . . . . . . . . . . . . . . . 26
3.1.1 Terms,TypesandFormulasinOCaml . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.4 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.5 SimpleTools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Zipperposition:aModularTheoremProver . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.2 Extensibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.3 LessonsLearntfromImplementingZipperposition . . . . . . . . . . . . . . 40
4 LinearIntegerArithmetic 42
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.2 NormalizationofLiteralsandClauses . . . . . . . . . . . . . . . . . . . . . . 46
4.1.3 PurificationofClauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 InferenceRules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.1 GroundVersionoftheRules. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.2 LiftingtoFirst-Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.1 SimplificationRules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.2 Subsumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.3 InequalityDemodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.4 SemanticTautologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
ii
4.4 VariableElimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6.1 RepresentationofLinearExpressions . . . . . . . . . . . . . . . . . . . . . . 62
4.6.2 MonadicIteratorsforBacktracking . . . . . . . . . . . . . . . . . . . . . . . 62
4.6.3 UnificationAlgorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6.4 OtherImplementationNotes . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6.5 GraphicalOutputforDebugging . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.7 ExperimentalEvaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 StructuralInduction 72
5.1 InductiveTypesandModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.1 NotationsandDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.2 RestrictionsontheTermOrdering . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.3 DealingwithConstructors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1.4 SemanticsandMinimalModels . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 InductiveStrengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 ProvingandUsingLemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3.1 GuessingLemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 InductiveStrengtheningusingSeveralClauses . . . . . . . . . . . . . . . . . . . . . 85
5.4.1 ExistenceofanInductiveModelforaSubsetofClauses . . . . . . . . . . . 86
5.4.2 EncodingtoQBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4.3 InferenceRulesandDependencyTracking . . . . . . . . . . . . . . . . . . . 88
5.4.4 SummaryofSpecialBooleanLiterals . . . . . . . . . . . . . . . . . . . . . . 90
5.4.5 InductiononOneConstant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4.6 InductiononSeveralConstants . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4.7 ExamplesandFurtherDiscussion . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5 ReconstructingProofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5.1 SATresolutionsproofsforInductiveStrengthening . . . . . . . . . . . . . . 97
5.5.2 QBFresolutionproofsusingUNSAT-cores . . . . . . . . . . . . . . . . . . . 98
5.6 ImplementationinZipperposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.6.1 InterfacingtoBooleanSolvers . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.6.2 ReducingtheQBFtoCNF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.6.3 ExperimentalEvaluationofZipperposition+Induction . . . . . . . . . . . . 102
6 TheoryDetection 104
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2 Higher-OrderReasoner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2.2 Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.3 CalculusfortheReasoner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2.4 Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4 ExperimentalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7 Conclusion 121
iii
Acknowledgements
First, I wish to thank my directors, the unordered set {Gilles,Guillaume}, for their kind sup-
port for the last three years — even longer for Gilles as he used to be my teacher years ago.
I also thank the reviewers of the thesis, for their careful reading of the manuscript and their
helpfulcommentsandfeedback—Ifeelprivilegedthattheydevotedsomeoftheirtimeand
expertisetoreviewingmywork—,andofcoursethejuryforacceptingtoattendmydefense.
Before I started my thesis, I had the chance to learn from people from Proval (in particular
Jean-ChristopheandAndrei),andSRI(Shankar,Sam,Bruno...).
(nowswitchingtofrench...)
Ensuite,jeremercieévidemmentmesparentsetmasœur,grâceauxquelsj’aiacquislegoût
delalectureetaieuunescolaritéheureusequiafinalementmenéàcettethèse;leursoutien
inconditionnel et leur aide ont été précieux pour la soutenance (merci aussi à mes oncles et
tantes). Pauline ailluminé letempsquejen’ai paspasséà(cyber-)scribouiller desarbresde
symboles. Merci également à mes amis du lycée (aux carrières artistiques ou médicales), de
laprépa(dédicacetrinômedecolle,bienévidemment: lecheveluetlemercenaire),etdel’X
(µ,Shuba,etlerestedugroupefameuxpoursonintelligencecollective;fufu,Anne,Nathaniel,
#bll,etbiend’autresgeeks;leurs+1,eeepc,etc.)
Deducteam dans son ensemble a été un endroit très agréable pour travailler et discuter
science(oulibre,ouautres),cegrâceauxautresdoctorants,stagiaires,etchercheurs.Enparti-
culier,Raphaëlaeulapatiencedem’écouterlorsdenombreusespausescafépost-prandiales;
PierreHalmagrandpoursondealershipefficacedecapsulesdubreuvagesus-mentionné; Ali
pourlestrollsetBaklavas;GuillaumeBurypourlesdiscussionstechniques,enparticuliercelle
oùilm’asauvéd’unecalvitiefulguranteliéeàunepreuvearithmétiqueépineuse;Davidpour
l’évocationdelasuper2position;Arnaudm’aencouragéàprocrastiner;ettouslesautres!
(andbacktoenglish...)
Last,IwanttothanktheauthorsofthevariousfreesoftwareIhavebeenusing.Inparticular,
in no specific order: LATEX(which allowed me to write this document and other papers), vim,
linux, git (for my peace of mind), OCaml, merlin, etc. TPTP and its nice maintainer, Geoff
Sutcliffe,werealsobothveryhelpful.
1
Chapter 1
Introduction
LogicandProofsinComputerScience
Logic,thelanguageandartofformalreasoning,isveryusefulinbothComputerScienceand
Mathematics. Bothrequirecorrect,unambiguousargumentationtosupportclaims; thecen-
tralconceptoftheorempreciselydesignatesaclaimbackedbysuchanirrefutableargument,
calledaproof. ThisfocusonformalproofsisquitecharacteristicofMathematics—anotable
exceptionistheprojectofLeibniztodesigna“calculusratiocinator”thatwouldmakeformal,
unambiguous reasoning the norm. However, proofs have a major drawback: it is in general
verydifficulttofindaprooftosupportagivenclaim. Humanexperts(usuallycalled“mathe-
maticians”or“logicians”)areundoubtedlythebestatfindingproofs; evenmoresowhenthe
problemisaboutfindingelegantproofs. Ontheotherhand,manytheoremsaretrulyboring.
Forinstance,theoremsgeneratedtoensurethatsoftwareabidesbysomespecificationarenei-
therelegantnorfuntoprove.Programsabletodischargeautomaticallythoseproofobligations
arethereforequiteusefulinpractice,eventhoughtheyprobablywillnotbeabletoprovehard
theorems(e.g.,theGoldbach’sconjecture)intheforeseeablefuture.Thestudyofprogramsthat
(tryto)provetheoremsisautomatedtheoremproving.
FromResolutiontoSuperposition
Automatedtheoremprovinghasbeenanactivefieldofresearcheversincethe1960s. Within
thisdiscipline,first-orderlogicplaysanimportantrôle,asitoccupiesasweetspotinthetrade-
offbetween having nice computational properties — as inthe case of propositional logic—,
andfeaturingahighlevelofexpressiveness—theclimaxbeingarguablyreachedbythehigher-
order,dependentlytypedlogicsusuallyfoundinproofassistantssuchasCoq[CDT].Focusing
onfirst-orderlogic,wecanadmireaquitediversifiedecosystemofcalculi;amongthem,Res-
olution[Rob65]anditsoffspring, Superposition[BG90,NR99]—whichaddsgoodreasoning
aboutequalityoveruninterpretedfunctions—,havebenefitedfromdecadesoftheoreticalim-
provementsandimplementationeffortsinvariouslanguages.Nowadays,Superposition-based
+
theoremprovers[RV01b,Sch02,WSH 07]areverycompetitiveinthefirst-ordercase.
Superpositionisnotenough
Eventhen,mereSuperpositionfallsshortformanyapplications:somemayrequiresomearith-
meticreasoning, somemaybeheavywithspecificalgebraictheories, somemayneedinduc-
tivereasoningtoreasononinductivestructures—inpractice,thoseaboundinprogramming,
Mathematics, etc. Extending Superposition has been an active research domain, going back
to handling the theory of Associative Commutative symbols — we might say it culminated
2
withtheproofoftheRobbinsconjecturebytheautomatedtheoremproverEQP[McC97]. Re-
cently,anextensioncalledAVATAR[Vor14]proposestointerfaceaSAT-solverwithinSuperposi-
tion,soastodelegatepropositionalreasoningtoit. Anotherextension,HierarchicSuperposi-
tion[BGW94,BW13],addsabackgroundtheorysolver—forinstancealineararithmeticsolver
—toSuperposition,inordertoreasonmodulothatparticulartheory.
Inthisthesis,weaimedatdevelopingnewextensionstoSuperposition. Ourclaimisthat
Superposition lends itself very well to being grafted additional inference rules and reasoning
mechanisms, mostlyremaininginaclausalsaturationframework. Saturation, thatSuperpo-
sitionshareswithitsancestor, Resolution, possessmany interestingpropertiesforreasoning
atthefirst-orderlevel(asopposedtoboolean-levelreasoning,foundinHierarchicSuperposi-
tion): whenaclauseisdeduced,itcanbeusedseveraltimes,makingproofsDAGsbysharing
sub-proofs;inaddition,usingfreevariables(implicitlyquantifiedattheclauselevel)leverages
unificationtoefficientlyfindrelevantinstancesofterms. Developingextensionsasdeductive
inferencesonfirst-orderclausesallowustodeducenewquantifiedtruthseveninthepresence
oftheories,orinamorepowerfullogic(inductivelogic).
TheimportanceofImplementation
Automated theorem proving is theoretically solved: the space of proofs is recursively enu-
merable, so a program that enumerates the possible proofs and checks whether they are a
proofofF isavalidproceduretotrytoproveF. Thismethodiscriticallyinefficient, forsev-
eral reasons: (i) it enumeratesall the uninteresting theorems, for instance every instance of
(A∧A∧...∧A)⇒(A∨A∨...∨A)for(m,n)∈(cid:78)+×(cid:78); (ii)itdoesnotusethegoaltoguideits
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
mtimes ntimes
search. Inpractice, decadesofresearchhavebeendedicatedtostudyingalgorithmsthatbe-
havelessstupidlyonactualtheorems.Thismakesautomatedtheoremprovingbothanexperi-
mentalandatheoreticaldomain. Ourworkisorientedtowardsprototypingandexperimenta-
tion;eachextensionwebuilthasitsownimplementationinZipperposition,aSuperposition-
based theorem prover developed for this very purpose. A chapter of the thesis is dedicated
topresentingZipperposition,itsimplementation,aswellasafoundationallogiclibrarycalled
Logtk.
OrganizationofthisThesis
Our main contributions, in addition to the pure implementation work, are threefold; conse-
quently,theyaredetailedinthreeseparatechapters(Chapters4,5,6).Theorganizationofthis
thesisis:
• In Chapter 2, preliminary mathematical and logic notions are defined, and their nota-
tions, introduced. The Superposition and AVATAR calculi are also presented. After this
chapter, the reader should have a clear idea of the notions required to understand the
nextchapters.
• Chapter3focusesontheimplementationpartofthethreeyearsweworkedonthisthe-
sis.ItstartsbypresentingLogtk,ageneral-purposeOCamllibraryforrepresentingtypes,
terms,formulas,etc. —notionsmathematicallydefinedinChapter2—inadditiontoa
collectionofclassicalgorithmssuchasunification,CNFtransformation,ortermindex-
ing. Then, Zipperposition, a theorem prover we built upon Logtk, is introduced. This
chapter is not a lecture in the implementation of automated theorem provers; it only
underlinessomeissuespertainingtowritingprogramsthatsearchforproofs.
• In Chapter 4, a Superposition-based calculus for integer linear arithmetic (also called
Presburgerarithmetic),inspiredbytheworkofWaldmannoncombiningSuperposition
withrationalarithmetic[Wal01]. LinearIntegerArithmeticisawidelystudiedandused
3
theory inotherareasofautomateddeduction, inparticular SMT(Satisfiability Modulo
Theory).
Lineararithmeticproblemsareoftenencounteredinautomatedproving,whetheritbe
directlyforprogramverification,orassertingcoherenceofcompileroptimizations. In-
deed,mostprogramsusebuilt-inarithmetic,andoftencanbeformalizedinlineararith-
metic.Acompilermight,forinstance,meetthefollowingsnippetofCcode:
for (i=1; i≤10; i++) a[j+i]=a[j];
The compiler may be interested in loading the value of a[j] once before entering the
loop, rather than loading it repeatedly inside the loop, since memory access is usually
slow. However, to ensure this optimization is safe, the compiler must assert that the
valueofa[j]doesnotchangewithintheloop. Onewaytodosoistoprovethatthere
isnoindexcollisionintheloop,whichcanbeformalizedbyprovingthearithmeticfor-
mula ∀i ∈(cid:90). 1≤i ≤10⇒ j (cid:54)= j +i. In addition to pure arithmetic reasoning or com-
putations (including program verification), other problems that have a discrete, totally
ordered structure, such as temporal logic, might be encoded into first-order logic with
arithmeticefficiently.
• InChapter5,wedefineanextensionofSuperposition+AVATAR thatisabletoreasonby
structuralinductiononnaturalnumbers, lists, binarytrees, etc. Inductionisattractive
because a local reasoning (prove that P(0) holds, and that if P is true on n, then P is
true on n+1) allows to prove universal properties (∀n :(cid:78). P(n): the property P holds
onallnaturalnumbers). Again,inductivereasoningisextremelyprevalentinComputer
Science,logic,andprogramming;yetthetworealmsoffirst-ordertheoremproversand
inductiveproversarestillmostlyseparate. Whereasspecializedinductiveproverssuch
asSpike[BKR92,Str12]areverysuccessfulinthelatter,theydonotshineintheformer.
Wetryheretobridgethegapfromtheoppositeside.
• Chapter6isdedicatedtoatheorydetectionsystemthat,givenasignature-agnosticde-
scriptionofalgebraictheories,detectstheirpresenceinsetsofformulas. Itsintegration
inZipperpositioncanalsodetectspecificinductivetheories(suchasthePeanoaxioms
fornaturalnumbers,whenpresentedasaninductivetype).Anearlyversionofthiswork
waspublishedin[BC13].
Eachchapterisrelativelyself-contained—commondefinitionsandtechniquesfromthestate
oftheartarefirstlistedinChapter2.Readersinterestedonlyinonechaptermightthenreadit
directlyafterChapter2.
4
Chapter 2
Technical Preliminaries
WestartwithagentleintroductiontothemathematicalconceptsandbasicsofLogicthatev-
erythingelseinthisthesisisbuiltontopof.Everythingtakesplaceinaclassicalsetting,mean-
ingtheprincipleofexcludedmiddle (foranyproposition p, p∨¬p holds)isavailableforthe
theoremprovertouse.
Remark2.1(DefinitionalEquality). Inthisthesis,ad=efbmeansthataisequaltobbydefinition
ofa.Weoftendefinenewvariablesthisway.
2.1 MathematicalConcepts
We use some very classic mathematical notions; in particular, we assume the reader knows
aboutsets. a ∈b meansthat a isamemberofthesetb. Setcomprehensionisnoted{x ∈a |
p(x)} — the set of all x that are members of a and satisfy property p —; the cardinal #s of a
finitesetsisthenumberofelementsitcontains.
Definition2.1(NaturalNumbers). Thenaturalnumbersarethepositiveintegers{0,1,2,...}.The
setofallnaturalnumbersisdenoted(cid:78),andthesetofstrictlypositivenaturalnumbersis(cid:78)+d=ef
(cid:78)\{0}.
Definition2.2(Integers). Thesetofintegers{...,−1,0,1,2,...}isdenoted(cid:90).
Definition2.3(Multiset). Amultisetisacollectionofobjects,likeaset,butinwhicheachitem
canoccurseveraltimes.Moreformally,amultisetisafunctionM fromasetS(calledthecarrier
ofM)to(cid:78);anelementx∈S hasmultiplicityi iffM(x)=i. Wesayx belongstoM,orx∈M,iff
M(x)≥1. Theunionoperator∪,definedby(M ∪M )(x)d=efM (x)+M (x),isoftenuseful. The
1 2 1 2
supportofamultisetM isthesubsetofS thathaveastrictlypositivemultiplicity. Wewillonly
considerfinitemultisets,thatis,multisetswhosesupportisafiniteset. Intherestofthisthesis,
wewillgenerallyuseset-likenotationsformultisets.
Definition2.4(Order). Anorderisabinaryrelation≤suchthatthefollowingaxiomshold:
Reflexivity :∀x.x≤x;
Transitivity :∀x y z.x≤y∧y≤z⇒x≤z;
Antisymmetry :∀x y.x≤y∧y≤x⇒x=y.
Definition2.5(StrictOrder). Astrictorderisabinaryrelation<satisfying:
Irreflexivity :∀x.x(cid:54)<x;
Transitivity :∀x y z.x<y∧y<z⇒x<z.
Definition2.6(Well-foundedOrder). Awell-foundedorderisastrictorder<suchthatthereis
noinfinitesequencex1,x2,...suchthat∀i.xi+1<xi. Wemightspeakofwell-foundedrelations
whentherelationadmitsnoinfinitesequence.
5
Definition2.7(PartialOrder,TotalOrder). Astrictorder<istotal(uptosomeequalityrelation
=)if, foranytwoobjects x,y, either x = y, or x < y, or y <x; otherwise, theorderispartial. A
non-strictorder≤istotalif,foranytwoobjectsx,y,eitherx≤y ory≤x;otherwiseitispartial.
Remark2.2(Termination). Sometimes,ifatransitiverelation→(intuitively,arewriterelation)
iswell-founded,wemightsayitisterminating.
Definition2.8(LexicographicCombination). Thelexicographiccombinationoftwostrictorder
< ,< ,alsonoted(< ,< ) ,isastrictorderdefinedonpairsby
1 2 1 2 lex
(cid:40)
x < y , or
(x ,x )(< ,< ) (y ,y )if 1 1 1
1 2 1 2 lex 1 2 x =y andx < y
1 1 2 2 2
Definition2.9(MultisetExtensionofanOrder). Themultisetextensionofanorder(cid:194),noted(cid:194)(cid:194),
isdefined[DM79]byM≺≺N iffthereexistmultisetsX,Y suchthat:
• X ⊆N;
• X (cid:54)=(cid:59);
• M=(N−X)∪Y;
• X dominatesY,thatis,∀y∈Y.∃x∈X.x(cid:194)y
whereM−N isthemultisetdefinedby(M−N)(x)=M(x)−N(x).
If(cid:194)iswell-founded,sois(cid:194)(cid:194);if(cid:194)istotal,sois(cid:194)(cid:194).
Definition2.10(TransitiveClosure). Thetransitiveclosureofabinaryrelation→,noted→+,is
definedinductivelyby
• ifu→v,thenu→+v;
• ifu→v andv→+w,thenu→+w.
Definition2.11(TransitiveReflexiveClosure). Thetransitivereflexiveclosureofabinaryrela-
tion→,noted→∗,isdefinedby
• foranyu,u→∗u;
• ifu→+v,thenu→∗v.
Definition2.12(Confluence). Arelation→isconfluentif,foranyx,y,zsuchthatx←∗y→∗z,
thereexistssomew suchthatx→∗w←∗z.
Definition2.13(NormalForm). Anobject x isanormalformforarelation→ifthereisno y
suchthatx→y.Wesayy isanormalformofxifx→∗y andy isanormalformfor→.
If→isconfluent,thereisatmostonenormalformforeachobject;if→isterminating,then
everyobjecthasanormalform.
Definition2.14(DirectedAcyclicGraph(DAG)). Adirectedacyclicgraph,orDAG,isacollection
ofverticesV andedgesE ⊆V ×V (eachedgeconnectstwovertices),suchthatthereisnocycle;
thatis,thereisnosequenceofedges{(v1,v2),(v2,v3),...,(vn−1,vn)}suchthatvn=v1.
Definition2.15(SmallerPartofaSet). WedenoteN≺x thesetofelementsofN thataresmaller
thanxw.r.t.thewell-foundedorder≺.
WedenoteN(cid:185)x thesetofelementsofN thataresmallerthanx w.r.t. theorder(cid:185)(including
xifx∈N).
2.2 BooleanLogic
Boolean logic (or propositional logic) is the simplest form of logic widely used. It is simple
enough that even computers can easily deal with it1. It deals with statements, or formulas,
builtontopofatomicpropositions(atoms)thatonlycarrya“true”or“false”value,withoutany
furtherstructure.
1DecisioninpropositionallogicisNP-complete,butthisisstillmuchbetterthansemi-decidability!
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