IntroductiontoNonMonotonicReasoningMasterRechercheSIS,Marseille ... - PDF document

IntroductiontoNonMonotonicReasoningMasterRechercheSIS,Marseille NicolaOlivetti ProfesseuralaFacult´eEcononomieAppliqu´ee,Universit´ePaulCezanneLaboratoireCNRSLSIS2010-2011a IamindebtedtoLauraGiordanoandAlbertoMartelliforhavingprovidedmetheircoursematerial.IntroductiontoNonMonotonicReasoning–p.1/36 Non-monotonicreasoning Oftenavailableknowledgeisincomplete. However,tomodelcommonsensereasoning,itisnecessarytobeabletojumptoplausibleconclusionsfromthegivenknowledge. Todrawplausibleconclusionsitisnecessarytomakeassumptions. Thechoiceofassumptionsisnotblind:mostoftheknowledgeontheworldisgivenbymeansofgeneralruleswhichspecifytypicalpropertiesofobjects.Forinstance,"birdsy"means:birdstypicallyy,buttherecanbeexceptionssuchaspenguins,ostriches,...IntroductiontoNonMonotonicReasoning–p.2/36 Non-monotonicreasoning Nonmonotonicreasoningdealswiththeproblemofderivingplausibleconclusions,butnotinfallible,fromaknowledgebase(asetofformulas). Sincetheconclusionsarenotcertain,itmustbepossibletoretractsomeofthemifnewinformationshowsthattheyarewrong Classicallogicisinadequatesinceitismonotonic:ifaformulaBisderivablefromasetofformulasS,thenBisalsoderivablefromanysupersetofS:S`BimpliesS[fAg`B,foranyformulaA.IntroductiontoNonMonotonicReasoning–p.3/36 Non-monotonicreasoning Example:lettheKBcontain:Typicallybirdsy.Penguinsdonoty.Tweetyisabird. ItisplausibletoconcludethatTweetyies. HoweverifthefollowinginformationisaddedtoKBTweetyisapenguinthepreviousconclusionmustberetractedand,instead,thenewconclusionthatTweetydoesnotywillhold.IntroductiontoNonMonotonicReasoning–p.4/36 Non-monotonicreasoning Thestatement"typicallyA"canbereadas:"intheabsenceofinformationtothecontrary,assumeA". Theproblemistodenetheprecisemeaningof"intheabsenceofinformationtothecontrary". Themeaningcouldbe:"thereisnothinginKBthatisinconsistentwithassumptionA". Otherinterpretationsarepossible Differentinterpretationsgiverisetodifferentnon-monotoniclogicsIntroductiontoNonMonotonicReasoning–p.5/36 InadequacyofClassicalLogic Wecannotrepresentarulesuchas"typicallybirdsy"as8x(bird(x)^:exception(x)fly(x))andthentoadd8x(exception(x)penguin(x)_ostrich(x)_canary(x)_:::) Wedonotknowinadvanceallexceptions Inordertoconcludethat"Tweety"'yweshouldprovethat"`tweetyisnotanexception"',thatis::penguin(tweety);:ostrich(tweety);:::IntroductiontoNonMonotonicReasoning–p.6/36 InadequacyofClassicalLogic OnthecontrarywewouldliketoprovethatTweetyiesbecausewecannotconcludethatitisanexception,notbecausewecanprovethatitisnotanexception.IntroductiontoNonMonotonicReasoning–p.7/36 ClosedWorldAssumption Abasicunderstandingofdatabaselogic,isthatonlypositiveinformationisrepresentedexplicitly.Negativeinformationisnotrepresentedexplicitly. Ifapositivefactisnotpresentinthedatabase(DB),itisassumedthatitsnegationholds. ThisiscalledClosedWorldAssumption:theonlytruefactsaretheprovableones. IfDB6`AthenDB`CWA:A Thisinferenceisnotvalidinclassicallogic.IntroductiontoNonMonotonicReasoning–p.8/36 ClosedWorldAssumption Example:supposeaDBcontainsfactsoftheform"practice(person,sport)"',forinstance:practice(anne;tennis)practice(joe;tennis)practice(anne;sky) ThenwehaveDB`CWA:practice(joe;sky) TriviallyCWAisnon-monotonic,sinceaddingafactmayleadtowithdrawthenegativeconclusion:DB[fpractice(joe;sky)g6`CWA:practice(joe;sky)IntroductiontoNonMonotonicReasoning–p.9/36 FrameProblem Problemofrepresentingadynamicworld Howtorepresentthatobjectsarenotaffectedbystatechange? Example:movinganobjectdoesnotchangeitscolor Inarepresentationbasedonaclassical-logic,wemustexplicitlyassertthepersistenceofobjectproperties.Weneedagreatnumberofframeaxioms,suchas:8x8c8s8l(color(x;c;s)color(x;c;result(move;x;l;s)))8x8c8s(color(x;c;s)color(x;c;result(t_light_on;s)))8x8c8s(color(x;c;s)color(x;c;result(open_door;s)))...IntroductiontoNonMonotonicReasoning–p.10/36 FrameProblem Wewouldneedageneralmeta-axiomoftheform:8p8a8s(holds(p;s)^:exception(p;a;s)holds(p;result(a;s))) Butthenwemustbeabletoconcludethatanactionisnotanexceptiontothepreservationofagivenproperty,unlesswecanshowthatitactuallyis. Weneedanon-monotonicreasoningmechanism.IntroductiontoNonMonotonicReasoning–p.11/36 NonMonotonicLogics Non-Monotoniclogicshavebeenproposedatthebeginningofthe80's,herearehistoricallythemostimportantproposals: Non-monotoniclogic,byMcDermottandDoyle,'80 DefaultLogic,byReiter,'80 Circumscription,byMcCarthy,'80 Autoepistemiclogic,Moore'84IntroductiontoNonMonotonicReasoning–p.12/36 DefaultLogic Defaultlogicextendsclassicallogicbynon-standardinferencerules.Theserulesallowsonetoexpressdefaultproperties. Example:bird(x):fly(x) fly(x)thatcanbeinterpretedas:"`ifxisabirdandwecanconsistentlyassumethatxiesthenwecaninferthatxies"'IntroductiontoNonMonotonicReasoning–p.13/36 DefaultLogic Moregenerallywecanhaverulesoftheform:(x):(x) \r(x)thatcanbeinterpretedas:"`if(x)holdsand(x)canbeconsistentlyassumedthenwecanconclude\r(x)". terminology: (x):theprerequisite (x):thejustication \r(x):theconsequentIntroductiontoNonMonotonicReasoning–p.14/36 DefaultTheory AdefaulttheoryisapairD;W&#x-278;&#x.571;,whereDisasetofdefaultrulesandWisasetofrst-orderformulas. Example:letletD;W&#x-278;&#x.209;beD=fbird(x):fly(x) fly(x)gW=fbird(tweety);8x(penguin(x)bird(x));8x(penguin(x)!:fly(x))gIntroductiontoNonMonotonicReasoning–p.15/36 DefaultTheory Intuitively,inadefaulttheoryD;W&#x-278;&#x.209;: Wrepresentsthestable(butincomplete)knowledgeoftheworld DrulesforextendingtheknowledgeWbyplausible(butdefeasible)conclusions. Notionofextensionofadefaulttheory:thetheory(=deductivelyclosedsetoflogicalformulas)obtainedbyextendingWbytherulesinD.IntroductiontoNonMonotonicReasoning–p.16/36 DefaultTheory Example:letD;W&#x-278;&#x.571;beasinthepreviousexample Sincebird(tweety)istrue,anditisconsistenttoassumefly(tweety),thenfly(tweety)istrueinthe(unique)extensionofD;W&#x-278;&#x.209;. ConsidernowthethedefaulttheoryD;W0&#x-278;&#x.209;,whereW0=W[fpenguin(tweety)gthentheassumptionfly(tweety)isnolongerconsistent,andtheapplicationofthedefaultruleisblocked.IntroductiontoNonMonotonicReasoning–p.17/36 DefaultTheory Example2:letD;W&#x-278;&#x.209;beasfollows:D=fd1=Rep(x)::Pac(x) :Pac(x);d2=Quack(x):Pac(x) Pac(x);gW=fRep(Nixon);Quack(Nixon)gForbothdefaultrulesdi,theprerequisiteisderivablefromW.WhatcanbeconcludedfromD;W&#x-278;&#x.571;? Ifweapplyd1,weconclude:Pac(Nixon);thereforePac(Nixon)cannotbeassumedconsistently,sothatd2isblocked. Ifweapplyd2,weconcludePac(Nixon);therefore:Pac(Nixon)cannotbeassumedconsistently,sothatd1isblocked.IntroductiontoNonMonotonicReasoning–p.18/36 DefaultTheory Therearetwoextensions:onecontaining:Pac(Nixon)andtheothercontainingPac(Nixon). Anextension(tobedenednext)representsthesetofplausibleconclusions. Asweshallsee,adefault-theorymayhavezero,one,ormanyextensions.IntroductiontoNonMonotonicReasoning–p.19/36 Extensions(propositionalcase) Givenadefaulttheory
=D;W&#x-278;&#x.209;,asetofformulasEisanextensionof
,if: Eisdeductivelyclosed:E=Th(E) allapplicabledefaultswithrespecttoEhavebeenapplied,thatisforall: \r2Dif2Eand:62Ethen\r2E Deductiveclosureoperator:Th(S)=fC2LjS`CgIntroductiontoNonMonotonicReasoning–p.20/36 Extensions:semi-inductivedenition Givenadefaulttheory
=D;W&#x-278;&#x.209;,asetofformulasEisanextensionof
,ifitcanbeobtainedasfollows: S0=W Si+1=Th(Si)[f\rj: \r2D;2Si;:62Eg E=SiSiIntroductiontoNonMonotonicReasoning–p.21/36 Extensions:semi-inductivedenition Thedenitionisnotreallyinductive,sincethedenitionofSi+1makesreferencetothewholeE. TheorderinwhichdefaultsareconsideredinstepSi+1issignicant:differentordersgiverisetodifferentextensions. Inthepropositionalcaseeveryextensioncanbe"generated"inatmostkstageswherekisthenumberofdefaultsinthedefaulttheory.IntroductiontoNonMonotonicReasoning–p.22/36 Extensions Example1:let
=fb;p!:fg;fb:f fg,thenthereisauniqueextensionE=Th(fb;p!:f;fg) S0=fb;p!:fg S1=S0[ffg,sinceS0`band:f62EIntroductiontoNonMonotonicReasoning–p.23/36 Extensions Example1':let
=fb;p!:f;pg;fb:f fg,thenthereisauniqueextensionE=Th(fb;p!:f;pg) S0=fb;p!:f;pg S1=S0,sinceS0`bbut:f2EIntroductiontoNonMonotonicReasoning–p.24/36 Extensions Example2:let
=fr;qg;fd1=r::p :p;d2=q:p pg. LetE1=Th(fr;q;:pg) S0=fr;qg S1=S0[f:pg,byapplyingd1,sinceS0`rand::p62E1 S2=S1,sinced2cannotbeapplied:p2E1 fori2,Si=S2IntroductiontoNonMonotonicReasoning–p.25/36 Extensions Example2(continued) LetE2=Th(fr;q;pg) S0=fr;qg S1=S0[fpg,byapplyingd2,sinceS0`qand:p62E2 S2=S1,sinced1cannotbeapplied:::p2E2 fori2,Si=S2IntroductiontoNonMonotonicReasoning–p.26/36 Extensions Example3Let
=W;D&#x-278;&#x.571;,whereW=;andD=f:a :ag.SupposethereisanextensionE if:a62E,thenitmustbe:a2E(wemustapplythedefault) butif:a2E,thedefaultbecomeinapplicable:thusitmustbe:a62E
hasnoextensions!IntroductiontoNonMonotonicReasoning–p.27/36 Extensions Example4Let
=W;D&#x-278;&#x.571;,whereW=;andD=fd1=::p q;d2=::q pg. LetE1=Th(fqg) S0=; S1=S[fqg,,since::p62E. S2=S1,sinced2becomesinapplicable. Similarly,wegetanotherextensionE2=Th(fpg)IntroductiontoNonMonotonicReasoning–p.28/36 Extensions Example4Let
=W;D&#x-278;&#x.571;,whereW=;andD=fa:b b;b:a ag.ThenthereisauniqueextensionE=Th(;) S0=; S1=S0,sinceS06`a,andS06`bIntroductiontoNonMonotonicReasoning–p.29/36 Normaldefaults Adefaultdisnormalifhastheform:  Anormaldefaulttheory
=W;D&#x-278;&#x.571;isadefaulttheorywherealldefaultsinDarenormal Theorem:Anormaldefaulttheoryhasalwaysanextension.IntroductiontoNonMonotonicReasoning–p.30/36 Inferencerelation Sinceadefaulttheory
=W;D&#x-278;&#x.209;mayhavemultipleextensions(includingnone),howtodeneanotionofinference?Therearetwonaturalnotions: (credulousinference)
`cAifthereexistsanextensionEof
suchthatA2E. (skepticalinference)
`sAifforallextensionsEof
,wehaveA2E. Sinceadefaulttheorymayhavenoextensions
`sAdoesnotimply
`cA.IntroductiontoNonMonotonicReasoning–p.31/36 Asimplealgorithm Analgorithmtocomputeanyextensionofatheory
=W;D&#x-278;&#x.571; (0)LetS0;D0&#x-278;&#x.209;=W;;&#x-278;&#x.209;.(i+1)LetX;Y;Z&#x-278;&#x.209;=Si;;;DDi&#x-278;&#x.571;foreveryd2Z,d=d:d \rdifSi[X`dandSi[X6`:dthenX;Y&#x-278;&#x.571;=X[f\rdg;Y[fdg&#x-278;&#x.209;letSi+1;Di+1&#x-278;&#x.209;=Si[X;Di[Y&#x-278;&#x.209; stopwiththeleastksuchthatSk;Dk&#x-278;&#x.209;=Sk+1;Dk+1&#x-278;&#x.209; checkwhetherforeachd=d:d \rd2Dk,Sk6`:d.If"yes",returnSk.IntroductiontoNonMonotonicReasoning–p.32/36 Problemswithdefaultlogic Unwantedtransitivity:let
=W;D&#x-278;&#x.209;,whereW=fstudentgandD=fd1=student:adult adult;d2=adult:works worksg itiseasytoseethat
hasauniqueextensionincludingfstudent;works;adultg. itisratherunintuitive(asstudentsusuallydonotwork). ifweaddthedefaultstudent::work :work,thetheoryhasthentwoextensions:E1=fstudent;adult;worksgE2=fstudent;adult;:worksg ButE2ismoreplausiblethanE1IntroductiontoNonMonotonicReasoning–p.33/36 Problemswithdefaultlogic Solution:replaced2by:adult:works^:student works thentheonlyextensionisE2=fstudent;adult;:worksg thisdefaultisnotnormal itissemi-normal:thejusticationimpliestheconsequent asemi-normaldefaulttheory(=atheorywherealldefaultaresemi-normal)mayhavenoextensionsIntroductiontoNonMonotonicReasoning–p.34/36 Problemswithdefaultlogic Handlingspecicity:let
=W;D&#x-278;&#x.571;,whereW=fuser;blacklistedgandD=fd1=user:login login;d2=user^blacklisted::login :loging thetheoryhasthentwoextensions:E1=fuser;blacklisted;logingE2=fuser;blacklisted;:loging ButofcourseonlyE2istheintendedone.IntroductiontoNonMonotonicReasoning–p.35/36 Problemswithdefaultlogic Theproblemofspecicitycanbehandledbyassigningaprioritytodefaultsonthebaseoftheirspecicity.Thepriorityorderistakenintoaccountforcalculatingextensions. Reiter'sDefaultlogichasalsootherproblems(e.g.cumulativity) Manyvariantshavebeenproposed,suchasBrewka'soneandLukaszewicz'sone.IntroductiontoNonMonotonicReasoning–p.36/36