Goal-Directed Proof Theory

273 Pages·2000·1.206 MB·English

by  Gabbay D., Olivetti N.

DOV GABBAY ANDNICOLA OLIVETTI
GOAL-DIRECTED PROOF THEORY
FOREWORD
Thisbookcontainsthebeginningsofourresearchprogramintogoal-directedde-
duction. The idea of presenting logics in a goal-directed way was conceived in
1984 when the first author was teaching logic at Imperialcollege, London. The
aimwastoformulatetheentireclassicallogicina goal-directedprolog-likeway
seethebook[Gabbay,1981]and[Gabbay,1998]. Thisformulationforbothclas-
sical and intuitionistic logic was successful and gave rise to an early version of
the Restart rule and N-Prolog, the first extension of Horn clause programming
with hypotheticalimplication (1984–1985). Since that time, severalauthorsand
researchgroupshaveadoptedthegoal-directedmethodologyeitherasafavourite
deductionsystemforsomespecificlogicorasafavouriteextensionofProlog.
Inthisbookwepresentgoal-directeddeductionasamethodologytobeapplied
tothemajorfamiliesofnon-classicallogics. Fromthepointofviewofautomated
deduction,whatmakesgoal-directednessadesirablepresentationisthatitdrasti-
callyreducesthenon-determinisminproofsearch.Moreover,agoal-directedpro-
cedureis by definition capableof focussing on the relevantdata for the proofof
thegoal,ignoringtherest. Evenifthismayseemaminorpointinsmalltheorem-
provingexamples, it is a most importantfeature if non-classicallogics are to be
usedtospecifydeductivedatabasesorlogicprograms. Whenevertheset ofdata
compriseshundredsof formulas,mostofwhichareirrelevanttothe proofofthe
currentgoal, one cannotjust randomlyselect a formulaof the data to processat
thenextstep. Heregoal-focusingorgoal-directednessmethodsbecomeessential.
Furthermore,asweshallseeinthisbook,theproofsystemswepresentarenot
just of interest in the pursue of automatization, but also for theoretical reasons.
Ourpresentationhighlightaproceduralinterpretationoflogicalsystems. Thereis
acommonnatureofalldeductivesystemsmadeofproceduralrules,anddifferent
proof systems can be seen as procedural variants, or perturbation, of the same
deductionalgorithm.
We do not try to give a general definition or pattern of goal-directedness, it
wouldberatherartificialasdefiningwhatisasequentcalculusoratableausystem
in general. Nevertheless, all sequent calculi or tableaux methods have a family
resemblancewhichjustifytheappellationofamethodasa‘tableaumethod’even
withoutaformaldefinitionofthisconcept.Thesameholdstruewithgoal-directed
proofmethodspresentedinthisbook.
Thisbookis only the beginningof ourresearchprogram. It demonstratesthe
thepossiblityofdevelopinggoal-directedproofproceduresforavarietyoflogical
iv GOAL-DIRECTEDPROOFTHEORY
systemsrangingoverintuitionistic, intermediate,modalandsubstructurallogics.
Ourfutureresearchtopicsarediscussedinthelastchapter.
We hopewe presentedenoughmaterialin this volumeto enable the readerto
applythismethodologytohisownfavouritelogicsystem.
Thebookisdirectedto researcher,undergraduateorgraduatestudentswithat
least an elementary knowledge of classical logic and some non-classical logics,
suchasmodallogics. Itcanbeusedin acourseonnon-classicallogicandauto-
mateddeduction,complementingstandardpresentationsofnon-classicallogicand
proof-theory.
DovM.Gabbay
NicolaOlivetti
London,February2000
Acknowledgements
WearegratefultoMatteoBaldoni,MarcelloD’Agostino,LauraGiordano,Alberto
Martelli, PierangeloMiglioli, DanieleMundici,AlessandraRusso, LucaVigano`,
SergeiVorobyovforvaluablediscussionsandcomments.
We are indebted to Agata Ciabattoni and Ulrich Endrissfor carefully reading
andcommentingonthemanuscript.
SpecialthanksareduetoBobMeyerandAlasdairUrquhartforvaluableexpla-
nationandinformationonrelevantlogics.
ThanksalsotoRoyDyckhoffandJamesHarlandforprovidinguswithrelevant
materialforthebook.
Thesecondauthorgratefullyacknowledgesthesupportandencouragementof
allhiscolleaguesoftheLogicProgrammingandAutomated-ReasoningGroupof
theUniversityofTorino. MoreoverheexpresshisgratitudetoHansTompitsand
thestaffoftheKnowledgeBasedSystemGroupatViennaUniversityofTechnol-
ogyforhavinginvitedhimto holdaone-termcourseonthesubjectofthisbook
in May 1999. He thanks the students who took part for fruitful discussion and
remarks.
Finally we would like to thank Mrs Jane Spurr, King’s College Publications
Manager,forherusualefficiency,dedicationandexcellenceinproducingthefinal
manuscript.
The second author was partially supported in this research by a six-month
fellowship from the Italian Consiglio Nazionale delle Ricerche, Comitato per le
Scienzematematiche,in1997.

CONTENTS
CHAPTER1 INTRODUCTION 1
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 ASURVEYOFGOAL-DIRECTEDMETHODS . . . . . . . . . . . 6
3 GOAL-DIRECTEDSYSTEMSARECUT-FREE . . . . . . . . . . . 14
4 ANOUTLINEOFTHEBOOK . . . . . . . . . . . . . . . . . . . . 15
5 NOTATIONANDBASICNOTIONS . . . . . . . . . . . . . . . . . 16
CHAPTER2 INTUITIONISTICANDCLASSICALLOGICS 21
1 ALTERNATIVEPRESENTATIONSOFINTUITIONISTICLOGIC . 21
2 RULESFORINTUITIONISTICIMPLICATION . . . . . . . . . . . 26
2.1 SoundnessandCompleteness . . . . . . . . . . . . . . . . . . . . 28
2.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 BOUNDEDRESOURCEDEDUCTIONFORIMPLICATION . . . 33
3.1 BoundedRestartRuleforIntuitionisticLogic . . . . . . . . . . . . 39
3.2 RestartRuleforClassicalLogic . . . . . . . . . . . . . . . . . . . 47
3.3 SomeEfficiencyConsideration . . . . . . . . . . . . . . . . . . . . 54
4 CONJUNCTIONANDNEGATION . . . . . . . . . . . . . . . . . . 58
5 DISJUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6 THE8;!-FRAGMENTOFINTUITIONISTICLOGIC . . . . . . . 78
CHAPTER3 INTERMEDIATELOGICS 93
1 LOGICSOFBOUNDEDHEIGHTKRIPKEMODELS . . . . . . . 94
2 DUMMETT–GO¨DELLOGICLC . . . . . . . . . . . . . . . . . . . 99
2.1 UnlabelledProcedurefortheImplicationalFragmentofLC . . . . 102
2.2 SoundnessandCompleteness . . . . . . . . . . . . . . . . . . . . 107
3 RELATIONWITHAVRON’SHYPERSEQUENTS. . . . . . . . . . 111
CHAPTER4 MODALLOGICSOFSTRICTIMPLICATION 117
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2 PROOFSYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3 ADMISSIBILITYOFCUT. . . . . . . . . . . . . . . . . . . . . . . 123
4 SOUNDNESSANDCOMPLETENESS . . . . . . . . . . . . . . . . 126
5 SIMPLIFICATIONFORSPECIFICSYSTEMS . . . . . . . . . . . . 132
5.1 SimplificationforK,K4,S4,KT:DatabasesasLists . . . . . . . . 133
5.2 SimplificationforK5,K45,S5:DatabasesasClusters . . . . . . . 138
0 GOAL-DIRECTEDPROOFTHEORY
6 ANINTUITIONISTICVERSIONOFK5,K45,S5,KB,KBT . . . . 142
7 EXTENDINGTHELANGUAGE . . . . . . . . . . . . . . . . . . . 146
7.1 Conjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2 ModalHarropFormulas . . . . . . . . . . . . . . . . . . . . . . . 148
7.3 ExtensiontotheWholePropositionalLanguage . . . . . . . . . . . 152
8 AFURTHERCASESTUDY:MODALLOGICG. . . . . . . . . . . 158
9 EXTENSIONTOHORNMODALLOGICS . . . . . . . . . . . . . 162
10 COMPARISONWITHOTHERWORK . . . . . . . . . . . . . . . . 164
CHAPTER5 SUBSTRUCTURALLOGICS 171
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
2 PROOFSYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
3 BASICPROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . 178
4 ADMISSIBILITYOFCUT. . . . . . . . . . . . . . . . . . . . . . . 181
5 SOUNDNESSANDCOMPLETENESS . . . . . . . . . . . . . . . . 192
5.1 SoundnesswithRespecttoFineSemantics . . . . . . . . . . . . . 195
6 EXTENDINGTHELANGUAGETOHARROPFORMULAS . . . . 202
6.1 Routley–MeyerSemantics,SoundnessandCompleteness . . . . . . 206
7 ADECISIONPROCEDUREFORIMPLICATIONALR . . . . . . . 220
8 AFURTHERCASESTUDY:THESYSTEMRM0 . . . . . . . . . . 231
9 RELATIONWITHOTHERAPPROACHES. . . . . . . . . . . . . . 244
CHAPTER6 CONCLUSIONSANDFURTHERWORK 249
1 AMOREGENERALVIEWOFOURWORK . . . . . . . . . . . . 249
2 FUTUREWORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
INDEX 265
CHAPTER1
INTRODUCTION
1 INTRODUCTION
This book is about goal-directed proof-theoretical formulations of non-classical
logics. It evolved from a response to the existence of two camps in the applied
logic(computerscience/artificialintelligence)community. Therearethosemem-
berswhobelievethatthenewnon-classicallogicsarethemostimportantforap-
plicationsandthatclassicallogicitselfisnolongerthemainworkhorseofapplied
logicandtherearethosewhomaintainthatclassicallogicistheonlylogicworth
consideringand that within classical logic the Horn clause fragmentis the most
important.
ThebookpresentsauniformProlog-likeformulationofthelandscapeofclas-
sicalandnon-classicallogics,doneinsuchawaythatthedistinctionsandmove-
mentsfromonelogic to anotherseem simple andnatural; and withinit classical
logic becomes just one among many. This should please the non-classical logic
camp. Itwillalsopleasetheclassicallogiccampsincethegoal-directedformula-
tionmakesitalllooklikeanalgorithmicextensionoflogicprogramming.
Thespectacularriseinnon-classicallogicanditsroleincomputerscienceand
artificial intelligence was fuelled by the fact that more and more computational
‘devices’ were needed to help the human at his work and satisfy his needs. To
makesucha‘device’moreeffectiveinanapplicationarea,alogicalmodelofthe
mainfeature ofhumanbehaviourin that area was needed. Thuslogicalanalysis
of human behaviour became part of applied computer science. Such study and
analysisofhumanactivityisnotnew. Philosophersandpurelogicianshavealso
beenmodellingsuchbehaviours,andinfact, havealreadyproducedmanyofthe
non-classical logics used in computer science. Typical examples are modal and
temporallogics. Theyhavebeenappliedextensivelyinphilosophyandlinguistics
aswellasincomputerscienceandartificialintelligence.
Thelandscapeofnon-classicallogicsapplicationsincomputerscienceandar-
tificialintelligenceiswideandvaried.Modalandtemporallogicshavebeenprof-
itablyappliedtoverificationandspecificationofconcurrentsystems[Mannaand
Pnueli, 1981; Pnueli, 1981]. In the area of artificial intelligence as well as dis-
tributedsystems,theproblemofreasoningaboutknowledge,beliefandactionhas
receivedmuchattention; modallogics[Turner,1985;HalpernandMoses, 1990]
have been seen to provide the formal language to represent this type of reason-
ing. Relevance logics have been applied in natural language understanding and
1
2 GOAL-DIRECTEDPROOFTHEORY
database updating [Martins and Shapiro, 1988]. Lambek’s logic [1958] and its
extensionsarecurrentlyusedfornaturallanguageprocessing. Anothersourceof
interestinnon-classicallogics,andinparticularintheso-calledsubstructurallog-
ics(withtheprominentcaseoflinearlogic[Girard,1987])hasoriginatedfromthe
functionalinterpretationprovidedby the Curry–Howard’sisomorphism between
formulasandtypesin functionallanguages(see [Hindley,1997]foranintroduc-
tionandreferencestherein,seealsoalso[Wansing,1990;GabbayanddeQueiroz,
1992]).
In parallel with the theoretical study of the logics mentioned above and their
applications, there has been a considerableamountof work on their automation.
The area of non-classical theorem-provingis growing rapidly, although it is yet
notasdevelopedasclassicaltheoremproving.
Althoughthereisawidevarietyoflogics,wecangrouptheexistingmethodolo-
giesforautomateddeductionintoafewcategories.Mostoftheideasandmethods
fornon-classicaldeductionhavebeenderivedfromtheirclassicalcounterpart.We
haveanalyticmethodssuchastableaux,systemsbasedonsequentcalculi,meth-
odswhichextendandreformulateclassicalresolution,translationbasedmethods,
andgoal-directedmethods.
Wecanroughlydistinguishtwoparadigmsofdeduction:human-orientedproof
methodsversusmachine-orientedproofmethods.Wecancalladeductionmethod
human-oriented if, in principle, the formal deduction follows closely the way a
humandoesit. Inotherwords,wecanunderstandhowadeductionproceedsand,
more precisely, how each intermediate step is related to the original deductive
query. With machine-orientedproofmethodsthisrequirementis notmandatory:
the originaldeductive task might be translated and encodedeven in another for-
malism. The intermediate steps of a computationmight have no directly visible
relationshipwiththeoriginalproblem.
According to this distinction, natural deduction, tableaux and sequent calculi
areexamplesofthehuman-orientedparadigm,whereasresolutionandtranslation-
basedmethodsarebetterseenasexamplesofthemachineorientedparadigm. In
particularresolutionmethodsrequireustotransformthequestion‘doesQfollow
from (cid:1)?’, into the question ‘is (cid:1)(cid:3) [fQ(cid:3)g consistent?’, where (cid:1)(cid:3) and Q(cid:3) are
preprocessednormalforms of (cid:1) andQ. Steppingfrom(cid:1) and Q to (cid:1)(cid:3) and Q(cid:3)
onemayloseinformationinvolvedintheoriginal(cid:1)andQ.Moreover,thenormal
formsmay be natural only from the machine implementationpoint of view, and
notsupportedbythehumanwayofreasoning.
Machine-oriented proof methods are more promising from the point of view
ofefficiencythanhuman-orientedproofmethods. Afterall,efficiency,uniformity
andreductionofthesearchspacewerethemainmotivationbehindtheintroduction
ofresolution.
Thebasicfeaturesofgoal-directedmethodsarethat: theyarehumanoriented
and they are a generalization of logic programming style of deduction. Goal-
directedmethodscanbeseenasanattempttofillthegapbetweenthetwoparadigms:
ontheonehand,theymaintaintheperspicuityofhuman-orientedproofmethods,
1.INTRODUCTION 3
ontheotherhand,theyarenottoofarfromanefficientimplementation. Thereis
anotherreasontobeinterestedingoal-directedproofsearch. Althoughwegener-
allyspeakaboutdeduction,thereareseveraldifferenttasks/problemswhichcanbe
qualifiedasdeductive. Thesedifferenttasksmightbetheoreticallyreducibleone
toeachother,butamethodoranalgorithmtosolveonedoesnotnecessarilyapply
successfullytoanother. Tomakethispointmoreconcrete,assumewearedealing
withagivenlogicalsystemL(sayclassical,orintuitionisticlogic,ormodallogic
S4),weusethesymbol‘tomeanboththeoremhoodandconsequencerelationin
thatlogic.Comparethefollowingproblems:
1. given a formula A we want to know whether ‘ A, that is whether A is a
theoremofL;
2. we are given a set Γ containing10,000formulasand a formulaA and we
wanttoknowwhetherΓ‘A;
3. we are given a set of formulas Γ and we are asked to generate all atomic
propositionswhichareentailedbyΓ;
4. WearegivenaformulaAandasetofformulasΓsuchthatΓ 6‘ A. Weare
askedtofindasetofatomicpropositionsS suchthatΓ[S ‘A.
The list of problems/tasks might continue. We call the first problem ‘theorem-
proving’. The second problem is close to deductive-database query answering.
The third problem/task may occur when we want to revise a knowledge-baseor
a state description as an effect of some new incoming information. The fourth
problem is involved in abductive reasoning and, in practice, one would impose
various constraints on possible solution sets S. It is not difficult to see that the
second,thirdandfourthproblemsare reducibleto thefirstone. An algorithmto
determinetheoremhoodcanbeusedtosolvetheotherproblemsaswell. Suppose
we haveVan efficient theorem prover P. Problem 2 can be reduced to check the
theorem Γ ! A. Thus, we can feed P with this huge formula, run it and get
an answer. However it might be that the formulasof Γ have a particular simple
formatand,mostimportantly,onlyaverysmallsubsetofthem(say10formulas)
arerelevanttoobtaintheproofofA. EvenifourtheoremproverPhasanoptimal
complexity in the size of the data (Γ+A), we would rather prefer a deduction
methodwhichis,inprinciple,capableofconcentratingonthedatainΓwhichare
relevanttotheproofofAandignoretherest.
The theorem prover can be usedVto solve also Problem 3: just enumerate all
atomicformulaspi,checkwhether Γ !pi,giveasoutputtheatomicformulas
forwhichtheanswerisyes. Itislikelythattherearebettermethodstoaccomplish
this task! For instance in the case that Γ is a set of Horn clauses, one can use a
bottom-upevaluation;moregenerally,onewouldtrytogeneratethissetincremen-
tallybyasaturationprocedure.
Thetheorem-provercanbeusedtosolveProblem4inanonV-deterministicway:
guess a set S and check by the theorem-proverwhether Γ^ S ! A. Again,