Proof Transformations from Search-oriented into Interaction-oriented ...

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Proof Transformations from Search-oriented into Interaction-oriented ...

approaches[Ahrendtetal.,98,Bjrneretal.,98].Otherapplicationsinclude

andvalidatebyahumanuser. checkbyamachineforthesamereasonsthattheyareeasiertounderstand intoaninteraction-orientedcalculusisanintermediatestep;and(2)checking proofcorrectness,becauseinteraction-orientedproofsaretypicallyeasierto (1)thetranslationofproofsintonaturallanguage,forwhichthetransformation

classifydierenttypesoftransformations.Thersttypearetranslationsbetween inherentlydierentcalculi|forexample,betweensemantictableauxandreso- nally,in[Section8]wedrawconclusionsfromourwork. Toplaceourworkinthegeneralcontextofprooftransformations,webriey Applicationsofourprooftransformationaredescribedin[Section7];and

calculi.Suchtransformationsbetweendierentcalculiarerelevantfortherela- [Eder,92]addressesthequestionofp-simulationbetweendierentrst-order lution[Wolf,94]orbetweennaturaldeductionandresolution[Pfenning,84].1

typeoftransformationsarenormalisationandoptimisationofproofswithinthe arewellsuitedforthesedierentpurposes[Miller,84,Pfenning,84].Asecond tionbetweenautomatedproofsearchandhumaninteractionincasethecalculi samecalculus(withoutchangingthesetofavailablerules).Afurthertypeof transformations|oneofthemostimportanteldsinprooftheory|istheelim- cutelimination.Inthecontextofpresentingautomaticallygeneratedproofsto inationorintroductionofrules(resp.theirapplications)suchas,forexample, humans,theintroductionofcuts(orlemmata[PfenningandNesmith,90])isof interest.

tailor-madeforautomatedsearchareeliminated. Onecalculuscanbeseenasarenementoftheother,butitisnotjustasubtionbetweendierentmembersofthesamefamilyofcalculi(semantictableaux).oderasuperset.Tomakeproofseasiertounderstandforhumans,renementsInthispaper,wepresentatransformationofadierentkind.ItisatranslaBelow,thetwomainclassesoftechniquesforimprovingtableaucalculiarede- 2.1EnhancementsandImprovementsofTableauCalculi 2TheTwoParadigms

toactuallybechanged;heuristicsandtechniquesfororganisingtheproofsearch scribedingeneral.Onlythosetechniquesareconsideredthatrequirethecalculus arenotdiscussedastheydonotaecttheformoftheconstructedproofsand, thus,donotaecttheirtransformation.

advantageofshorterproofsandthedisadvantagethattheseshortproofsmay proceedateachtableauexpansionstep.Thus,thereisatrade-obetweenthe Strengtheningthecalculus.Acalculusisstrengthenedifitischangedinsucha eningaddsnon-determinismtoacalculus,i.e.,therearemorepossibilitiestowaythatshorterproofsforatleastsomeformulaeexist.Inmostcases,strength- behardertondastherearemorechoicepointsinthesearchspace. 1[Pfenning,84]actuallyusesexpansiontrees,whichcanbeseenasanabstraction expansiontreesintonaturaldeduction(butintheframeworkofhigherorderlogic).

ofresolutionproofs.[Miller,84]describesatranslationintheotherdirection,from


Post-poningchoicepoints.Often,itispossibletouniformlyrepresentdierent

dummy)variableisusedasaplaceholderforthedierentterms. termsbyothertermsareallrepresentedbyasingletableauinwhichafree(or [seeSection4.1],wheretableauxthatareidenticaluptothereplacementof tableaux|andthusdierentpartsofthesearchspace|byasingletableauusing additionalsyntacticaldevices.Atypicalexampleistherigidvariabletechnique

not(fully)automatedorthataproofiscommunicatedtotheuser. ahumanuser,wherethereasonforinteractionmayeitherbethatthesystemis 2.2Interaction-orientedCalculi Interaction-orientedcalculiareusedifadeductionsystemhastointeractwith

applicationeasytounderstand.2Inparticular,ruleapplicationsshouldnothave tactically(notnecessarilysemantically)simple,i.e.,eachapplicationshouldbe anynon-localsideeects,ashumanstendtomodulariseaproblemandfocus easytovalidate,thepre-conditionsshouldbeeasytocheck,andeectsofarule Humansprefercalculiwithsimplerules.Ruleapplicationsshouldbesyn-

onprovingonesub-problematatime. problemdomaineliminatestheneedtominimisethesearchspace.Highlynondeterministicrulesmaybeused|aslongastheyaresimpleinthesensethat thatimposesnorestrictionsonthecutformulaethatcanbeintroduced. theirapplicationsareeasytovalidate;anexampleisthenon-analyticcutrule Ontheotherhand,ahumanuser'sintuitionandmeta-knowledgeaboutthe

example,theinstantiationofrigidvariables). morecomplicatedinferencerulesandhavenon-localsideeects(suchas,for interaction-orientedcalculiastheytypicallyresultinrulesthataresyntactically Techniquesforpost-poningchoicepointsareoftennotsuitableforenhancing

2.3Search-orientedCalculi Forautomateddeduction,minimisingthesizeofthesearchspaceisthemain

increasesitsnon-determinism,weakerversionswithfewerchoicepointsareoften ofasmallersearchspace.Consequently,becausestrengtheningacalculusoften objectiveindesigningacalculus.Toachievethisgoal,non-determinismiselim- longproofs,thepossibilityofndingshortproofsmaybesacricedforthesake preferableforautomateddeduction.3 inatedwheneverpossible.Thoughshortproofsareingeneraleasiertondthan

todeducetheconclusion\or:"fromtheemptypremissforallformulae, 2Calculiforinteractivehigherordertheoremprovingoftenhaverulesthatarenot simpleinthatsense.Thereasonwhytheserulesareacceptableforahumanuser| despitetheircomplexity|isthattheyconstitutecomparativelylargeproofsteps. Strengtheningthecalculusbyaddinganon-analyticcutrule,whichallows

3Insearch-orientedtableaucalculi,techniquesarefrequentlyusedthatrestrictthe partofthelogicaresimple. Buteveninthesehigher-ordercalculi,atleasttherulesforhandlingtherst-order searchspacewithoutaectingtheformoftheconstructedproofs|suchasordering andconnectednessconditions[BeckertandHahnle,98,Letz,98].Theysacricethe possibilitytondshorterproofsforthesakeoflessnon-determinism.Suchtechniques arenotconsideredhere,asusingthemmeanschangingtheproofsearchwithout changingthecalculusitself.


estrictedversionsofthecutrule,suchasthegenerationoflocallemmata[see weighstheadvantageoftheexistenceofshorterproofs.Therefore,inautomated deductionsystemsthecutruleisneverusedunrestrictedly.Thereare,however, isatypicalexamplewherethedisadvantageofadditionalnon-determinismout- Section4.3],thatstillcanreducethesizeofthesmallestproofsexponentially toconsiderablysmallerproofs,andaddsonlyveryfewadditionalchoicepoints. butdonotleadtotoomanyadditionalchoicepoints.

searchiftheonlymeasureconsideredisthenumberofexpansionstepsthat istheintroductionofuniversalvariables[seeSection4.2];thistechniquecanlead Techniquesthatpostponechoicepointsarealwaysofadvantageforproof Amethodforstrengtheningtableaucalculithatisveryusefulforproofsearch

syntacticaldevicesandbookkeepingmechanismsarerequired,whichcanbe diculttoimplementandleadtocomputationaloverhead. havetobeexecutedtondatableauproof.Thedisadvantageisthatadditional

cisionaboutwhichpartofthesearchspaceshouldbefurtherinvestigated.Foradditionalinformationhasbeengatheredthatallowstomakeaninformedde- example,ifclosingatableaubranchrequiresarigidvariableXtobeinstantiated wheredierentpartsofthesearchspaceareinvestigatedsimultaneously,until Inacertainsense,post-poningchoicepointsleadstoabreadth-rstsearch

withacertaintermt,thenthedecisiontoinstantiateXwithtisinformedin whichisnotpossiblewithouttheinstantiation. thesensethattheinstantiationisknowntobeusefulasabranchclosureexists

rigidvariables). search-orientedcalculitohavemanyspecialisedinferencerules,(syntactically) complexrules,andruleswithnon-localsideeects(suchastheinstantiationof Notethat,incontrarytointeraction-orientedcalculi,itisacceptablefor

Inthissection,wedeneagroundtableaucalculusforrst-orderpredicatelogic, whichisatypicalrepresentativeoftheclassofinteraction-orientedcalculi.This versionoftableauxiscalledgroundbecauseuniversallyquantiedvariablesare 3AnInteraction-orientedTableauCalculus

sophisticated-ruleforhandlingexistentialquantication(seebelow),weuse ismeanttobeusedinteractivelyandnotautomatically,anunrestrictedcutrule instantiatedbygroundtermswhenthe-ruleisapplied.Exceptforamore isincluded. thestandardgroundtableaurules(asdenedby[Smullyan,68]).Asthecalculus

missesandconclusionsareseparatedbyahorizontalbar,whileverticalbarsin andtoformulapatterns. tationgivenin[Tab.1],wheretheleft-mostcolumnsassignthetypes,,,[Tab.2]containstheruleschemataforourinteraction-orientedcalculus(preThedenitionofthetableauexpansionrulesmakesuseoftheunifyingnoicalforaninteraction-orientedcalculusandallowstheusertomakeuseofhistheconclusiondenotedierentextensions).Theright-mostschemaisthenonanalyticcutrule,inwhichcanbeanyformula.easytounderstand.Moreover,theycanbeseenasdeningthecalculuscom- intuitionanddomainknowledge. Boththe-ruleandthecutrulearehighlynon-deterministic,whichistyp- Theexpansionruleschematain[Tab.2]allhaveaverysimpleformandare


:(1!2)1:2 :(1_2):1:2 1^2 :: 112 2 1_2 12

1(x) :(1^2):1:2 (1!2):12 1 2

:(9x)((x)):(x) (8x)((x))(x) :(8x)((x)):(x) (9x)((x))(x) 1(x)

12 1 2 Table1:Unifyingnotation.

Table2:Ruleschemataforthegroundversionoftableaux. wheretisany groundterm. 1(t) wherec=sko(). 1(c) :

thatisimplicitlyalreadycontainedin[Tab.2]. tion1merelyformalisesthedenitionoftableauxforagivensetofformulae pletely;noadditionalexplanationsareneeded(inparticular,closingabranch

Denition1.Atableauisanitelybranchingtreewhosenodesarerst-order doesnotchangethetableau).Thismakesthecalculusveryuser-friendly.Deni-

sentences4,thetableauxforare(recursively)denedby: formulae.AbranchinatableauTisamaximalpathinT.Givenasetof 2.LetTbeatableaufor,BabranchofT,and 1.Thetreeconsistingoftheonebranchisatableaufor(initialisation).5

for(-,-,-,-expansion). branchesarelabelledwiththeformulaeintheextensions,thenT0isatableau expansionrulecorrespondingto T0isconstructedbyextendingBbyasmanynewbranchesasthetableau hasextensions,wherethenodesofthe aformulainB.Ifthetree

3.LetTbeatableaufor,BabranchofT,andanarbitraryformula.If thetreeT0isconstructedbyextendingBbytwonewsinglenodebranches, onecontainingtheformulaandtheothercontainingtheformula:,then

itsbranchesareclosed. Denition2.GivenatableauT,abranchBofTisclosediBcontainsapair ;:ofcomplementaryformulae;otherwiseitisopen.Atableauisclosedifall T0isatableaufor(cut-expansion).

consistsofatableauTforthatisclosed. Denition3.Atableauprooffor(theunsatisabilityof)asetofsentences 4Asentenceisarst-orderformulanotcontaininganyfreevariables. 5Weidentifythesetwithabranchwhosenodesaretheformulaein.


olsforanygivenrst-ordersignatureofSkolemconstantsymbols,whichisdisjointfromthesetFoffunctionsym- -ruleschema.Forskolemisation,weusesymbolsfromaspecialinnitesetFsko Sofarwehavenotexplainedthemeaningoftheoperatorskousedinthe

Denition4.Givenasignature=hP;Fi,thefunctionskoassignstoeach hP;F[Fskoiisdenotedby -formulaover asymbolsko()2Fskosuchthatsko()>cforallc2Fsko . =hP;Fi.Theextendedsignature

occurringin,where>isanarbitrarybutxedorderingonFsko.

applicationoftheclassical(ground)-ruleisalwaysneww.r.t.thecontext.The intuitivereasonwhyour-ruleissoundalthoughtheSkolemsymbolcmaynot tableaubranch)whereour-ruleisappliedtoreplaceanexistentiallyquantiedvariablewithc.Incontrasttothat,theSkolemsymbolintroducedbyan Note,thatthesymbolc=sko()mayalreadyoccurinthecontext(onthe

benewisthefollowing:Ingeneral,a(ground)-ruleissoundifitguarantees thedependencyrelationbetweenSkolemsymbolsisacyclic.Thatdependency relationisthetransitiveclosureoftherelation;denedby:ForallSkolem symbolscandd,c;dic=sko('(d)).Aneasy(thoughnotoptimal)wayto ensurethatthedependencyrelationisacyclic,istodemandthateachSkolem symbolisneww.r.t.thecontextwhereitisusedbythe-rule.Ourimproved acyclic. -ruleismoreliberalbutensuresneverthelessthatthedependencyrelationis

bythefactthatourruleisstrictlylocal(whereastheconceptofa\new"symbolisinherentlyglobal).Anadditionalbenetofusinganimprovedruleinticallysimpleastheclassicalrule;thatdisadvantageis,however,madeupforbuttheyareusefulforgroundcalculiaswell.Our-rulemaynotbeassyntac- Suchmoreliberalruleswererstusedinfreevariablecalculi[seeSection4.1],

4ASearch-orientedTableauCalculus thegroundcalculusaswellisthatthetransformationbetweenthefreevariable (search-oriented)andtheground(interaction-oriented)calculusgetseasier.

automateddeductionwereferto[BeckertandHahnle,98]. mata.Foranoverviewoftheseandotheradvancedtechniquesfortableau-basedsearch.Weusetheconceptsofrigidvariables,universalvariables,andlocallem- Here,weintroduceatableaucalculusthatisdesignedforautomatedproof

whentheyareintroduced,theycanberepresentedbyarigidvariableX.Later, conclusionsoftheform1(t)foralltermst.Therefore,insteadofguessingterms Theconceptofrigidvariables6isbasedonthefactthatthe-ruleallowstoderive 4.1RigidVariables

whentheproofprocedurehasgoodreasontouseaparticularinstancet,the variableXisinstantiated\ondemand";thatmeans,alloccurrencesofXare replacedbyt.Usually,thisisdonewhenatleastonebranchcanbeclosed(in 6Intheliterature|inparticularoncalculiforrst-orderpredicatelogic|,several parameter,dummyvariable,andmetavariable.

othernameshavebeenusedforfreeand,inparticular,forrigidvariables,including


thesenseofDenition2)providedthatXisreplacedbyt.Unicationisused proofandthusthesizeofthesearchspace. variablesreducesthenumberofchoicepointsintheconstructionofatableau tondamostgeneralsubstitutionthatallowstocloseabranch.Usingrigid

bythesameterm(whichiswhythesevariablesarecalled\rigid"). someterm.Alloccurrencesofarigidvariableinatableauhavetobeinstantiated unknown)elementofthesetofallformulaethataretheresultofreplacingXby ThepossibilitytoinstantiaterigidvariableswithSkolemsymbolsleadsto Intuitively,aformulacontainingarigidvariableXstandsforasingle(but

dicultiesifa-ruleisusedwhosesoundnessisbasedondemandingtheSkolem symbolthatisintroducedtobenew.Consider,forexample,thetableaushownon tableaubecomesground(shownontherightin[Fig.1].Thisgroundtableau demandsSkolemsymbolstobenew.Butitcanbeconstructedusingtheground cannotbeconstructedusingagroundcalculuswiththeclassical-rulethat theSkolemsymbolc).7WhentheclosingsubstitutionfX=cgisapplied,the theleftin[Fig.1].Thelastexpansionruleappliedhasbeena-rule(introducing

calculusdenedin[Section3],whichhasamoreliberal-rule.Thisexample illustrateswhyatableauprooftransformationfromarigidvariablecalculusinto agroundcalculususingaliberal-ruleiseasierthanatransformationintoa calculususingtheclassical-rule. (8x)(p(x)^(9y):p(y)) p(X)^(9y):p(y) (9y):p(y) :p(c) p(X) (8x)(p(x)^(9y):p(y)) p(c)^(9y):p(y) (9y):p(y) :p(c) p(c)

Figure1:Atableauthatcannotbeconstructedusingtheclassicalground-rule. closedbyfX=cg

unknownterm,itcanbeusedaswelltorepresentallterms.Then,aformula eningatableaucalculus.InsteadofusingafreevariabletorepresentasinglebutUndercertainconditions,thereisanalternativeuseoffreevariablesforstrength- 4.2UniversalVariables

containingsuchafreevariableXstandsforthesetofallformulaethatarethe resultofreplacingXbysometerm.Intuitively,thesefreevariablescanbeseen universalvariables.Theadvantageofusinguniversalvariablesisthefollowing: asbeinguniversallyquantiedonthemeta-level;accordinglytheyarecalled 7Forthisexample,itisnotimportantexactlywhich-ruleisused.Itcanbethe Fitting,96]leadtothesameresult(however,therulegivenintherstedition rulewedenebelow,butearlierversionsofthe-rule[HahnleandSchmitt,94, of[Fitting,96]leadstoadierenttableau).


Oftenseveraldierentinstancesofatableauformulacontainingfreevariables themechanismtodosoistoapplythe-rulemorethanoncetothesamepre- havetobeusedtocloseabranch(orasub-tableau).Inrigidvariabletableaux, miss,introducingnewrigidvariablestogeneratevariantsofaformula.Suppose atableaubranchBcontainsaformula(X).WhenXisinstantiatedinthe tableau(tocloseabranch),then(X)isinstantiatedaswell.Inparticularsituations,however,itispossibletoprotectXin(X)frombeinginstantiated. Thisissoundwhenever(t)isalogicalconsequenceoftheformulaeonBfor alltermst.Thisisundecidableingeneral,butmanyoftheprotectablefree variablescanberecognisedeasily.Forthispurposeweintroducethenotionof decoratedformulae.

setofallfreevariablesin). variablesinFree()nUarecalledtherigidvariablesof(whereFree()isthe Denition5.AdecoratedformulaU:consistsofasetUoffreevariablesand aformula.ThevariablesinUarecalledtheuniversalvariablesof;andthe

Usingtheconceptofuniversalvariablesyieldsshortertableauproofs,andin mostcasesreducesthesearchspace.Ifbothuniversalandrigidvariablesareused thatX2U;theycanallbeusedwithoutrstgeneratingvariantsof(X). formulae.Intuitively,U:(X)representsthedierentinstances(t)provided Inthesearch-orientedcalculusdenedbelow,thetableaunodesaredecorated

inacalculus,thenclosingsubstitutionsareonlyappliedtotherigidoccurrences ofavariable.

offormulae.Thetoplefttreein[Fig.2]isarigidvariabletableauforthat Example1.Weconsidertableauxfortheset=f(8x)(p(x));:p(a)_:p(b)g thatdemonstratetheusefulnessofuniversalvariables. Beforedeningoursearch-orientedcalculusformally,wegivesimpleexamples

(toprightinthegure)canbededuced. bothbranches.Tondaproof,theexpansionrulehastobeappliedagaintothe cannotbeclosedimmediatelyasnosinglesubstitutionforXallowstoclose -formula(8x)(p(x))toaddavariantp(X0)ofp(X).Then,theclosedtableau

representsallformulaeoftheformp(t),includingp(a)andp(b). (bottomrightinthegure)withoutapplyingasubstitution,becausefXg:p(X) fXg:p(X)insteadofp(X).Thistableaucanbeexpandedtoaclosedtableau universalvariablesareusedinthisexample).Itcontainsthedecoratedformula Thebottomlefttableauinthegureisamixedvariabletableaufor(only

arefourdierentpossibilitiestoclosethebranch.If,however,thebranchcontains tableaubranchcontainstheformulaep(X1);p(X2);:p(a),and:p(b),thenthere thedecoratedformulafXg:p(X)insteadofp(X1)andp(X2),thenthereisonly redundanciesinherenttorigidvariablecalculi.If,forexample,arigidvariable Anadditionaladvantageofuniversalvariablesisthattheyhelptoavoid

4.3LocalLemmata 92];ithaslaterbeenimproved[Beckert,98,BeckertandHahnle,98]. onepossibility,namelyclosingthebranchwithoutinstantiatinganyvariable. Theconceptofuniversalvariableswasrstdescribedin[BeckertandHahnle,

tomakesurethattheextensionsofaconclusiondonotintersectsemantically.

Asimpleandinmanycasesusefulwayofstrengtheningatableaucalculusis


:p(a) (:p(a)_:p(b)) (8x)(p(x)) p(X) :p(b) :p(a) (:p(a)_:p(b)) (8x)(p(x)) p(X)

withfX=agandfX0=bgapplied p(X0) :p(b)

;::p(a) ;:(:p(a)_:p(b)) ;:(8x)(p(x)) fXg:p(X) ;::p(b) ;:(:p(a)_:p(b)) ;:(8x)(p(x))

Figure2:TableauxforExample1 ;::p(a) fXg:p(X) ;::p(b)

intersectsemantically,aspandqcanbothbesatisedbyasingleinterpretation. Forexample,thetwoextensionspandqoftheconclusionforthepremissp_q

pretation,thensuchaninterpretationhastobeconsideredandexcludedmoreofitsbranches.Iftherearebranchesthatmaybesatisedbythesameinterintersectingextensionsadds,insomesense,lessinformationtothetableau.Intuitively,tocloseatableau,onehastoshowthatnointerpretationsatisesany Theapplicationofanexpansionruleusingaconclusionwithsemantically

thismayrequiretheuseofadditionalextensions,andonehastobecarefulto riveconclusionsthatarenotintersection-free.Analternative-ruleschemais preservesoundness. thanonce.Extensionscanbemadeintersection-freebyaddingtableauformulae; The-ruleoftheinteraction-orientedcalculusfrom[Section3]allowstode-

whichproducesintersection-freeconclusions(theformula:1thatisaddedto therightextensioncanbeconsideredtobealocallemma).Usingthisnew 12

schemaleadstoanon-elementaryreductioninthesizeoftheshortestproofsfor :1

certainclassesofformulae[Egly,98]. itisnotincludedinoursearch-orientedcalculus. isalsoanintersection-freerule.However,forreasonsdiscussedin[Section2.3], Theunrestrictedcutruleoftheinteraction-orientedcalculus[seeSection3]

4.4TheCalculus Wenowhaveeverythingathandtodeneoursearch-orientedtableaucalculus. Itisamixedvariablecalculus,i.e.,itisworkingwithbothrigidanduniversal


locallemmata.Theruleschemataaregivenin[Tab.3].Thesquarebracketsin Thecalculusalsousesthe-ruleschemafrom[Section4.3],whichgenerates the-ruleschemaindicatethataddingthelocallemmaisoptional. variables.Forthispurpose,weusethedecoratedformulaefromDenition5.

U\Free(1):1 U\Free(2):2 U: ;:1;:2 U: [;::1] whereXisanew U[fXg:1(X) freevariable. U: fX1;:::;Xng=Free(). wheref=sko()and U:1(f(X1;:::;Xn)) U:

mustbechangedslightly.Now,FskoisaninnitesetofSkolemfunctionsymbols. Forthefreevariableversionofthe-rule,thedenitionoftheoperatorsko Table3:Ruleschemataforthemixedvariableversionoftableaux.

and(b)forall-formulae;0over allf2Fskooccurringin,where>isanarbitrarybutxedorderingonFsko, Denition6.Givenasignature=hP;Fi,thefunctionskoassignstoeach ticalifandonlyifand0areidenticaluptorenamingoffreevariables. -formulaover afunctionsymbolsko()2Fskosuchthat(a)sko()>ffor

thatresultfromunifyingpairsofpotentiallycomplementaryformulae.Universal Inmixedvariabletableaux,rigidvariablesareinstantiatedbysubstitutions ,thesymbolssko()andsko(0)areiden-

containingthesameuniversalvariablesareunied.However,insteadofactually variablesareprotectedfrombeinginstantiated,reectingthefactthatevery

formulaeappropriately. universalvariablescanberenamedtoavoidoccurcheckclasheswhenformulae renaminguniversalvariablesintableaux,wedenetheunicationofdecorated possibleinstanceofthecorrespondingformulacouldbederived.Accordingly,

inatableauTifitistherestrictionofasubstitutionwiththeproperty Denition7.AsubstitutionisaunierofdecoratedformulaeU:andU0:0 ()=(0)tothesetFree()nU)[(Free(0)nU0ofvariableswhereis arenamingofthevariablesinUandisarenamingofthevariablesinU0with

tableauTisamaximalpathinT.Givenasetofsentencesover,thetableaux variablesnewtoT.

forare(recursively)denedby: Denition8.Let branchingtreewhosenodesaredecoratedformulaeover bearst-ordersignature.Atableau(over)isanitely

1.Thesinglebranchconsistingofthedecoratedformulaeinf;:j2gisa .Abranchina

2.LetTbeatableaufor,BabranchofT,andU: inB.IfthetreeT0isconstructedbyextendingBbyasmanynewbranches tableaufor(initialisation).

extensions,thenT0isatableaufor(expansion).

asthetableauexpansionrulecorrespondingtoU: thenodesofthebranchesarelabeledwiththedecoratedformulaeinthe hasextensions,where adecoratedformula


3.LetTbeatableaufor,BabranchofT,andU: inB.IfisaunierofU: isatableaufor(closure). T0=T,i.e.,T0isconstructedbyapplyingtoallformulaeinT,thenT0 andU0: 0(accordingtoDenition7),and andU0:: 0literals

5TheTransformationofFreeVariableInstantiations

thetechniqueforavoidingtheintroductionofredundantproofpartsisdescribed whilepreservingtheoriginalproofstructureasmuchaspossible.Inthissection tainingadditionalredundancies.Itisdesirabletominimisetheseredundancies thatispartofourtransformation. Atransformationfromonecalculusintoanotheroftenresultsinaproofcon-

from[Section4]intotheinteractionrepresentedcalculusfrom[Section3]isthat groundcalculuswithoutfreevariables. theformerusesfreevariables(bothrigidanduniversal),whereasthelatterisa Themaindicultyintransformingproofsfromthesearch-orientedcalculus

tobederivedfrom(8x)((x))usingtheground-rule,andsubsequently formula: isderivedapplyingthe-ruletotheformula(8x)((x))andthatlaterasuccessorformula Assume,forexample,thatinafreevariabletableauprooftheformula(X) (a).Then,inthecorrespondinggroundproof,theformula(a)has (X)of(X)isusedtocloseabranchBunifyingitwithsome

becauseuniversalvariablescanbeinstantiatedmultiply.Mechanismsforex- toB.Ofcourse,thesituationisingeneralmuchmorecomplex|inparticular hastobederivedfrom(a)mimickingthederivationof thennallyallowstoclosethebranchinthegroundtableauthatcorresponds (X)from(X),which (a)

tensivebookkeepingareneededtokeeptrackofhowformulaearederivedand

inaformulahavebeenintroducedby-ruleapplicationshastobetakeninto groundversionofaproof,theorderinwhichdierentfreevariablesoccurring theinstance(a)ofthepredecessorformulahastobederivedrst). whichinstancesareneeded(toderiveaninstance Inaddition,tominimisethenumberofformulaethataregeneratedinthe (a)inthegroundcalculus,

account.Thisorderinducesadependencyrelationbetweenfreevariables,asthe followingexampleillustrates(asimilarsituationisdescribedinExample4):

theformula Example2.Assumethattwoinstancesp(a;b)andp(a;c)ofsomedecoratedfor-

instancesp(a;b)andp(a;c).Inthatcase,onecanmakeuseofthefactthatthe mulafX;Yg:p(X;Y)areusedinafreevariabletableauprooftoclosebranches. i.e.,ifXhasbeenintroducedrst,theninthegroundcalculusonecanderive Ifthisformulahasbeenderivedfromthe-formula=(8x)(8y)(p(x;y)),

onceto. groundinstancescoincideintheinstantiationofXbyapplyingthe-ruleonly If,however,the-formulaisoftheform0=(8y)(8x)(p(x;y)),i.e.,ifYhas =(8y)(p(a;y))andwithtwo-ruleapplicationsto theground

toderiveboth beenintroducedrstandXdependsonY,theninthegroundcalculusonehas numberofgeneratedformulae.

twogroundinstancescanbederived.Inthatcase,thefactthattheground instancescoincideintheinstantiationofXdoesnotleadtoareductioninthe 01=(8x)(p(x;b))and 02=(8x)(p(x;c)),fromwhichthenthe


inherenttothetreesreectsthedependencyrelationbetweenthefreevariables (theinstantiationsofthevariableintroducedrstarecontainedintherootsof thetrees).Theinstantiationtreescorrespondingtotheinstantiationsdescribed instantiationsofthefreevariablesoccurringinaformula.Theorderthatis Inthefollowing,weuse(setsof)instantiationtreestorepresent(setsof)

inExample2aredepictedin[Fig.3]. XYabac (a) XY(b) bc a YXbaca

theseinstantiationtrees.(c)Asetofinstantiationtreethatcannotbemerged(see Example2). Figure3:(a)Asetofinstantiationtreethatcanbemerged.(b)Theresultofmerging (c)

thatinstantiationsofsomedecoratedformula=fX;Y;Zg:p(X;Y;Z)have alreadybeencollectedandarerepresentedbytheinstantiationtreeshownin Example3.Inthisexample,weconsideraslightlymorecomplicatedcasewhere instantiationtreesaremergedthatconsistofmorethanonebranch.Assume

in[Fig.4(b)].Theresultofthemergingprocessisshownin[Fig.4(c)].Inthe [Fig.4(a)].Assumefurtherthattheinstancep(a;b;c)islaterusedtoclosea groundtableauproof,atotalnumberofsix-ruleapplicationsisneededto tableaubranch,i.e.,theinstantiationtreehastobemergedwiththeoneshown generatethethreerequiredgroundinstancesof(anaivetransformationwould generatenine-ruleapplications). XYZ bdef a XYZabc XYZ cdef ba

Figure4:Theinstantiationtree(c)istheresultofmergingthetrees(a)and(b)(Example3). (a) (b) (c)

ofthetableau-basedtheoremprover3TAP,worksinthreesteps: 1.Computeandcollectallinstantiationsoffreevariables(bothrigidanduni- Ourtransformationalgorithm,whichhasbeenimplementedinPrologaspart versal)requiredtocloseallbranchesofthegivenfreevariabletableauproof;


2.Computethenecessaryinstantiationsofthefreevariablesinallformulae,in- inthisrststeponlytheinstantiationsofvariablesoccurringinliterals

iscomputedbymergingthesetsofinstantiationtreesofallsuccessorforcludingthenon-literalformulae.Thesetofinstantiationtreesofaformularesentedbyinstantiationtreesattachedtotheseliterals.actuallyusedforbranchclosureareconsidered.Theinstantiationsarerep- 3.Constructthegroundproof.Itcontainsoneformulaforeachpathinany oftheinstantiationtreescomputedinStep2(itmaycontainadditional redundantformulae).Inaddition,atthisstage,thecomplexSkolemterms mulaeof.ispolynomialinthesizeoftheoriginalfreevariableproof.However,sincetheincreaseinproofsizeheavilydependsonthenestinglevelofuniversalquantiersin

Thesizeofthegroundtableauproofthatistheresultofthistransformation ofthefreevariableproofarereplacedbySkolemconstants.

Adierenttransformationthatmakesuseofthecutrulewassuggestedby[Egly, thegroundcalculus|totranslatemultipleinstantiationsofuniversalvariables. 98].Itcanleadtoshortergroundproofs(whichdependingonthepurposeof thatourtransformationdoesnotmakeuseofthecutrule|whichisavailablein theproof,theincreaseinproofsizeisinpracticeonlyslightlysuper-linear.Note

forourpurpose. muchaspossibleforthebenetofahumanuser,Egly'smethodisnotsuitable incertainenvironments,butasouraimwastopreservetheproofstructureas oftheproofdrastically.Suchachangeoftheproofstructuremaybetolerable theprooftransformationcanbeimportant);however,itchangesthestructure

Insteadwegivethreeexamplestoillustratethedicultiesarisingwiththetrans- inmoredetailhere(adetaileddescription[inGerman]isavailable[Stenz,97]). formationandhowtheyaresolvedbyourmethod. Duetospacelimitationswecannotdescribethetransformationalgorithm

p(a;d),andp(c;d),whichareusedtoclosethethreebranchesofthetableau thetransformation,i.e.,instantiationtreeshavebeencomputedforallliterals literalp(X;Y)aredepictedinthegure;theycorrespondtoitsinstancesp(a;b), Example4.[Fig.5(a)]showsafreevariabletableauproofaftertherststepof

(p(X;Y)canbeusedwithdierentinstantiationsforthevariablesXandYas usedtocloseoneofthebranches.Onlytheinstantiationtreesofthenon-ground

arenotshownforgroundformulae(forwhichthelistisalwaysempty). andthefollowinggures,thelistsoffreevariablesbeinguniversalinaformula emptyinstantiationtreeisimplicitlyattachedtogroundliterals).Also,inthis theyareuniversalinthisliteral).Allotherliteralsinthistableauareground(the

tree.Note,thatitisnotpossibletomergethetwotreeswiththeidenticalleafd astheyhaveincompatiblerootsaandc.Inaddition,thesetofinstantiation treesforthenon-literalformula(8y)(p(X;y))hasbeencomputed. Thetwoinstantiationtreesofp(X;Y)withrootahavebeenmergedintoone Theresultofthesecondstepofthetransformationisshownin[Fig.5(b)].

isshown.Thedottedarrowsindicatethecorrespondencebetweenthepathsof theinstantiationtreesin[Fig.5(b)]andthegroundformulaein[Fig.5(c)].

In[Fig.5(c)]thenalresultofthetransformation(agroundtableauproof)


fXg:(8y)(p(X;y)) (8x)(8y)(p(x;y))

fX=a;Y=bg:(p(a;b)^p(a;d)) :((p(a;b)^p(a;d))^p(c;d)) fX;Yg:p(X;Y) XYabadcd

:p(a;b) :p(a;d) (a) :p(c;d) fX=a;Y=dg fX=c;Y=dg

fXg:(8y)(p(X;y)) (8x)(8y)(p(x;y))

:((p(a;b)^p(a;d))^p(c;d)) fX;Yg:p(X;Y)

:p(a;b) :(p(a;b)^p(a;d)) :p(a;d) :p(c;d)

XYa Xac (8x)(8y)(p(x;y)) (8y)(p(a;y))

bdcd (8y)(p(c;y)) p(a;d) p(a;b)

:(p(a;b)^p(a;d)) :((p(a;b)^p(a;d))^p(c;d)) p(c;d)

(b) :p(a;b) :p(a;d) (c) :p(c;d)

Figure5:Anexampleforthetransformationwhereinstantiationtreescontainonly

tratethedicultiesarisingifinstantiationtreescontainnon-groundterms.The groundterms(Example4).

(notethatthisisdierentfromtheemptyinstantiationtree). symboloccurringininstantiationtreesrepresentsanarbitraryinstantiation Example5.Thefreevariabletableauproofsshownin[Figs.7(a)and8(a)]illus-

f(Y)ofthevariableXcaneasilybemerged,becausethevariableYhastobe maindierenceisthatintherstcase[seeFig.7]thetwoinstantiationsf(a)and instantiatedwithaanyway.Inthesecondcase[seeFig.8],however,thereisno singleoptimalresultofthetransformation.Inthiscase,theinstantiationsf(a) Thefreevariabletableauproofsinthesetwoguresareverysimilar.The

andf(Y)ofXarenotmergedbyouralgorithm.Itgeneratestheinstanceq(b) ofq(Y)andtheinstancesp(f(a))andp(f(b))ofp(X)[seeFigure8(b)].The resultofthetransformationusingtheseinstantiationsisshownin[Fig.8(c)]. casethetwoinstantiationsofXcanbemergedandgeneratingthesinglein-

Onecouldaswellgeneratethetwoinstancesq(a)andq(b)ofq(Y),inwhich


stancep(f(a))ofp(X)issucient.Then,theresultinggroundproofwouldbe dierentfromtheoneshownin[Fig.8(c)]butitwouldbeofthesamesize. ingexampledemonstrates,ourtransformationdoesnotalwaysndanoptimal solution(forwhichallinstantiationsinthewholetableauwouldhavetobe ductionofredundanciesintotheresultinggroundproof.However,asthefollow consideredsimultaneously). Themainreasonforintroducinginstantiationtreeswastoavoidtheintro-

Example6.AssumethattwoinstancesofadecoratedformulafXg:p(X)have beenusedforbranchclosure:p(a)andp(Y)whereYisafreevariablealso occurringonsomeotherbranchofthetableau.Ourtransformationdoesnot withatocloseotherbranchesofthetableau(thissituationisshownin[Fig.6]. mergethetwoinstantiationsofXalthoughthatisusefulifYisinstantiated

p(a) fXg:p(X) p(Y) XaY :q(a) q(Y)

Example7.Theproblemillustratedbythenalexampleisthatclosingatableau Figure6:ThesituationdescribedinExample6.

ingY.ThatdoesnotcauseproblemsiftheuniversalvariabletechniqueisusedbranchmayrequiretoinstantiateafreevariableYwithatermg(Y)contain- (andYisuniversal).However,specialcarehastobetakenwhenafreevariable aSkolemfunctionsymbolforwhichg=sko()where=:(8x)(f(Y)f(x)). tableaucontainingsuchaninstantiationistransformedintoagroundtableau; arisesinthefreevariabletableauproofshownin[Fig.9(a)].Inthisproof,gis theinstantiationhastobesplitintotwoseparateinstantiations.Thissituation

havetobegeneratedinthegroundversionoftheproof. tion,theconstantchasbeenchosentobethearbitrarytermthatisrepresented byin[Fig.9(a)].Note,thattwoinstantiationtreesareattachedtoeachof thenon-literalformulae,indicatingthattwoseparateinstancesoftheseformulae In[Fig.9(b)],whichshowstheresultofthesecondstepofthetransformastancesofthe-formulaThegroundproof,whichisshownin[Fig.9(c)],containstwodierentin-

termg(g(c))inthefreevariableproof).Sincee\depends"ond,wehavee>d ableproof)andthesecondconstantise=sko( constantsisd=sko( accordingly,twodierentSkolemconstantsareintroduced.Therstofthese (c))(whichcorrespondstothetermg(c)inthefreevari- (Y)=:(8x)(f(Y)f(x)),namely (d))(whichcorrespondstothe (c)and (d);

where>istheorderingonFskothatensuresthedependencyrelationonSkolem ina-ruleapplication,thisproofcouldnotbeconstructedusingtheclassical constantstobeacyclic(Def.4). ground-rulethatdemandsSkolemconstantstobenew.

SincedalreadyoccursintheproofatthetimeitisusedasaSkolemconstant


fYg:q(Y)^:(p(f(a))^p(f(Y))) (8y)(q(y)^:(p(f(a))^p(f(y))))

fY=agfYg::(p(f(a))^p(f(Y))) :q(a)_(8x)(p(x)) fYg:q(Y) Ya

fX=f(a)g :q(a) :p(f(a)) fXg:p(X) (8x)(p(x))

(a) ;::p(f(Y)) Xf(a)f(Y) Y fX=f(Y)g

fYg:q(Y)^:(p(f(a))^p(f(Y))) (8y)(q(y)^:(p(f(a))^p(f(y))))

fYg::(p(f(a))^p(f(Y))) :q(a)_(8x)(p(x)) fYg:q(Y) Ya

:q(a) :p(f(a)) fXg:p(X) (8x)(p(x)) Ya

;::p(f(Y))

Ya (8y)(q(y)^:(p(f(a))^p(f(y)))) q(a)^:(p(f(a))^p(f(a))) :(p(f(a))^p(f(a))) q(a)

Xf(a)Ya :q(a) :q(a)_(8x)(p(x))

:p(f(a)) (8x)(p(x)) p(f(a))

Figure7:Therstexampleforthetransformationwhereinstantiationtreescontain (b) (c) :p(f(a))

freevariables(Example5). tion7.2].Atranslationofthegroundproofintonaturallanguageisshownin[Sec- 6TheTransformationofLocalLemmata Applicationsofthe-ruleinthesearch-orientedcalculus[seeTab.3],whichmay the-ruleofthetargetcalculus[seeTab.2]doesnotgeneratelemmata. availableinthetargetcalculus.A-ruleapplicationthatgeneratesalocallemma

introducelocallemmata,requireattentionwhentheyaretransformed,because cutruleapplicationandcanthereforeeasilybesimulatedsincethecutruleis Fortunately,theintroductionoflocallemmatacanbeseenasarestricted


fYg:q(Y)^:(p(f(a))^p(f(Y))) (8y)(q(y)^:(p(f(a))^p(f(y))))

fY=bgfYg::(p(f(a))^p(f(Y))) :q(b)_(8x)(p(x)) fYg:q(Y) Yb

fX=f(a)g :q(b) :p(f(a)) fXg:p(X) (8x)(p(x))

(a) ;::p(f(Y)) Xf(a)f(Y) Y fX=f(Y)g

fYg:q(Y)^:(p(f(a))^p(f(Y))) (8y)(q(y)^:(p(f(a))^p(f(y))))

fYg::(p(f(a))^p(f(Y))) :q(b)_(8x)(p(x)) fYg:q(Y) Yb

:q(b) fXg:p(X) (8x)(p(x)) Yb

:p(f(a)) ;::p(f(Y))

Yb (8y)(q(y)^:(p(f(a))^p(f(y)))) q(b)^:(p(f(a))^p(f(b))) :(p(f(a))^p(f(b))) q(b)

Xf(a)f(b) :q(b) :q(b)_(8x)(p(x)) (8x)(p(x))

Yb :p(f(a)) p(f(a)) p(f(b))

Figure8:Thesecondexampleforthetransformationwhereinstantiationtreescontain (b) (c) :p(f(b))

resultsinaredundantbranchthatcanimmediatelybeclosed);see[Fig.10]for isreplacedbyacutruleapplicationandasubsequent-ruleapplication(which freevariables(Example5).

anexample.

oftheavailablerules.Thetranslationoflocallemmata(asdescribedabove)is describeinwhichwayapplicationsofrulesthatdonotexistinthetargetcalculus aretobereplacedbynewsub-proofsconsistingofoneorseveralapplications simplelanguagefordeningtranslationpatternsismadeavailable,thatallowsto Intheimplementationofourtransformationaspartoftheprover3TAP,a

anapplicationofthisexiblemechanism.Anotherexampleisthesimulationof specialrulesin3TAP'ssearch-orientedcalculusthat,forexample,allowtoderive fromthepremissfalse_theconclusioninasinglestep.


pp_q:p q p p_q Cut p:p Figure10:Simulatingtheintroductionofalocallemmausingthecutrule. redundantbranch Closureoftheq

theproofcanbepresentedtotheuserand,secondly,KIVcanintegratetheresultsof3TAPintoitsextensiveproofmanagementsystem,thuseliminatingalot ofredundantwork.Theautomatedtheoremprover3TAPusesasearch-oriented sagessuchas\success"or"nosuccess".Thishastwomainadvantages.Firstly,goalwastopassproofsfoundby3TAPbacktoKIV,insteadofonlysimplemes- fragmentofwhichissimilartotheinteraction-orientedcalculuspresentedin [Section3]. interactivesystemKIVusesasequentcalculusfordynamiclogic,therst-order calculusthatisnearlyidenticaltotheonedescribedin[Section4],whilethe

Automated Theorem ProverDeductionSystem Interactive Theorem

orientedSearch- Prover

Calculus Interaction- User

Figure11:Theintegrationofautomatedandinteractivetheoremproving. Calculus oriented

7.2ProofPresentationinNaturalLanguage Anotherintegrateddeductionsystem(asshownin[Fig.11]hasbeendeveloped ontheotherhand;3TAPisoneoftheproversabletocooperatewithILF. designedasagenericinteractiveproverenvironmentforintegratingtheabilities ofhumanusersontheonehandandanysortofautomateddeductionsystem intheframeworkoftheILFproject[Dahnetal.,97].TheILFsystemhasbeen

theprooftransformationdescribedinthispaperpermitstheoutputofproofs featuresofILF[DahnandWolf,96,Wolf,98].Therefore,theimplementationof foundby3TAPtobetranslatedbyILFintonaturallanguage.

targetcalculusofourtransformationintonaturallanguageisoneofthemain Theabilitytotranslateproofsfromacalculussuchastheinteraction-oriented


Example8.ILF'stranslationofthegroundtableauprooffromExample7[see Fig.9]intonaturallanguageisshownin[Fig.12]. Proof.Weshowindirectlythat Letusassumethat(9y)(8x)(f(y)f(x))doesnothold: (9y)(8x)(f(y)f(x)) (1)

Wechoseanarbitraryconstantc1foryin(2)andobtainthat(8x)(f(c1) f(x))doesnothold:Hencewecanintroducec2forx.Thisgivesthat f(c1)f(c2)doesnothold: (2)

f(c1):Thereforewehaveacontradiction.Thuswehavecompletedtheproof assumethatf(c3)f(c1)doesnothold:Hencef(c1)doesnothold:By(3) Wechoseanarbitraryconstantc3foryin(2)andobtainthat(8x)(f(c3) f(x))doesnothold:Hencewecanreplacexbythearbitrarilychosenc1and (3)

Figure12:AtableauprooftranslatedbyILFintonaturallanguage(Example8). of(1).

theformuladoesnotdependonc1(thedependencybetweenSkolemconstantsis sen"whenitwasrstusedandbecause(2)theotherconstantc3occurringinalthoughc1alreadyoccurs.Thatisallowedbecause(1)c1was\arbitrarilycho- (implicitly)existentiallyquantiedvariablexin\f(c3)f(x)doesnothold" Notethat,inthelastparagraphoftheproof,theconstantc1replacesthe

discussedinSection3belowDenition4).The(satised)dependencycondition

7.3ProofChecking naturallanguageproofpresentation). isnotpointedoutbyILF(itisarguablewhetheritshouldbementionedina

Afurtherapplicationforprooftransformationsisthepossibilitytocheckthe correctnessofproofsmoreeasily. constructedmaynotbesoundinallcases,anditishighlydesirabletobeable cedures,theirimplementationsareverycomplex.Therefore,theproofsthatare search-orientedcalculioftenemploytableaurulesthatarediculttovalidate tocheckthecorrectnessoftheseproofs.Thisismadedicultbythefactthat Sinceautomatedtheoremproversusecomplicatedheuristicsandproofprople;forthesamereasonsthatproofsininteraction-orientedcalculusareeasier

tounderstandandvalidateforhumans[seeSection2.2],theyareeasiertocheck foracomputer.Theproofcheckercanbeimplementedasasmallandsimple andthathavenon-localsideeects[seeSection2.3].

programandisthereforemoretrustworthy.Therefore,theprooftransformation Therulesofaninteraction-orientedcalculus,however,aresyntacticallysim-

procedureisusefulforgeneratingtheinputforaproofchecker.


ugsinboth3TAPandtheimplementationoftheprooftransformation. implementedacompactproofchecker,whichhasbeenveryhelpfulindetecting Basedontheimplementationofourprooftransformationaspartof3TAP,we

8Conclusion Wehavecategorisedlogiccalculiintotwotypesaccordingtothepurposethey serveintheoremproving,eitherproofsearchoruserinteraction.Furthermorewe

groundcalculus,andwehaveestablishedaprooftransformationprocedurefrom haveexplainedtheinherentcharacteristicsofsuchcalculiandhowtheserelate

proofsintheformerintoproofsinthelattercalculus. categories,asearch-orientedfreevariablecalculusandaninteraction-oriented tothedesignatedr^oleofacalculus.

Webelievethattheproblemsweencounteredindevisingsuchatransfor- Asexampleswehavedenedonetableaucalculusfromeachofthosetwo

mationprocedurearenotparticulartoourcalculibutareinvariablycausedby theneedtotranslatebetweenacalculusplacingthegreatestemphasisonsearch

secondsteptheemptyclauseisderived. itsrulesasthemainobjective.Similarphenomenamay,forexample,occurifa resolutionproofistransformedintoagroundresolutionproof,i.e.,aproofwhere spaceminimisationandacalculusdesignedwiththesyntacticalsimplicityof rstasetofgroundinstancesoftheinputclausesarelisted,fromwhichina

considerableeort. betweencalculithatservedierentpurposeseveniftheybelongtothesame family.Bridgingthisgapandthusbringingproofsbacktotheuserrequires Suchtransformationsarenotonlyoftheoreticalvaluebuttheyhaveappli- Thisworkexempliesandsupportsthefactthatthereisanon-trivialgap

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Acknowledgements

ThisworkwassupportedbytheDeutscheForschungsgemeinschaft(DFG)within

theSchwerpunktprogrammDeduktion.WewouldliketothankIngoDahn,Uwe

Egly,andReinerHahnleforfruitfuldiscussions,anonymousrefereesforuseful

commentsonanearlierversionofthispaper,andThomasHonigmannforhis

helpinimplementinganinterfacetotheILFsystem.

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