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IES 503 ARTIFICIAL INTELLIGENCE

IES 503 ARTIFICIAL INTELLIGENCE. INFERENCE IN FIRST-ORDER LOGIC. İPEK SÜĞÜT. CONTENTS. PROPOSITIONAL VS. FIRST-ORDER LOGIC INFERENCE RULES FOR QUANTIFIERS UNIFICATION AND LIFTING FORWARD CHAINING BACKWARD CHAINING RESOLUTION CONCLUSION.

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IES 503 ARTIFICIAL INTELLIGENCE

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  1. IES 503 ARTIFICIAL INTELLIGENCE INFERENCE IN FIRST-ORDER LOGIC İPEK SÜĞÜT

  2. CONTENTS • PROPOSITIONAL VS. FIRST-ORDER LOGIC • INFERENCE RULES FOR QUANTIFIERS • UNIFICATION AND LIFTING • FORWARD CHAINING • BACKWARD CHAINING • RESOLUTION • CONCLUSION

  3. PROPOSITIONAL VS. FIRST-ORDER LOGIC INFERENCE RULES FOR QUANTIFIERS Withthehelp of theserules, we can converttheknowledgebasetopropositionallogicandusepropositionalinference in ordertousefirst-orderinference. Let us beginwithuniversalquantifiers. KB => standardfolkloricaxiomstatingthatallgreedykingsareevil: Then it is quiteeasytoinferany of thefollowingsentences:

  4. INFERENCE RULES FOR QUANTIFIERS • Universal Instantiation(UI):Infer any sentence obtained by substituting a ground term for the variable. Let denote the result of applying the substitution to the sentence . The rule is: for any variable and ground term . EXAMPLE: The three sentences given earlier are obtained with the substitutions • Existential Instantiation(EI):Morecomplicatedthan UI. Foranysentence variable , andconstantsymbolthatdoes not appearelsewhere in theknowledgebase:

  5. INFERENCE RULES FOR QUANTIFIERS EXAMPLE: => as long as does not appearelsewhere in the KB. *Theexistentialsentencesaysthere is someobjectsatisfying a condition, andtheinstantiationprocess is justgiving a name tothatobject. Mathematicsprovide a nice example:Supposewediscoverthatthere is a numberthat is littlebiggerthan 2.71828 andthatsatisfiestheequationforWe can givethisnumber a name, such as , but it would be mistaketogive it the name of an existingobject, such as . Inlogic, thenew name is called a Skolemconstantand EI is a specialcase of a more general processcalledskolemization.

  6. REDUCTION TO PROPOSITIONAL INFERENCE • Just as an existentiallyquantifiedsentence can be replacedbyoneinstantiation, a universallyquantifiedsentence can be replacedbythe set of allpossibleinstantiations. EXAMPLE:Our KB containsjustthesentences: Then, weapply UI tothefirstsentenceusingallpossiblegroundtermsubstitutions: Weobtain : Thistechnique is calledpropositionalization. There is a problem withthistechniqueifwehave a functionsymbol in KB. Because it gives us an infinite set of possiblegroundtermsubstitutionssuch as

  7. UNIFICATION AND LIFTING A first-orderinferencerule:Supposethatinstead of knowingweknowthateveryone is greedy: Then, wewouldstillliketo be abletoconcludethatbecauseweknowthat John is a kingand John is greedybecauseeveryone is greedy. Thisinferenceprocess can be captured as a singleinferencerulethatwecallGeneralizedModusPonens: Foratomicsentences, wherethere is a substitution 8 suchthat, foralli, EXAMPLE:

  8. UNIFICATION AND LIFTING *GeneralizedModusPonens is a liftedversion of ModusPonens (it raisesModusPonensfrompropositionaltofirst-orderlogic). *Thekeyadvantage of liftedinferencerulesoverpropositionalization is thattheymakeonlythosesubstitutionswhicharerequiredtoallowparticularinferencestoproceed. Unification:Liftedinferencerulesrequirefindingsubstitutionsthatmakedifferentlogicalexpressionslookidentical. Thisprocess is calledunificationand is a keycomponent of allfirst-orderinferencealgorithms. Thealgorithmtakessentencesandreturns a unifierforthemifoneexists: EXAMPLE:Supposewehave a query Someanswers can be foundbyfindingallsentences in KB thatunifywith. Here aretheresults of unificationwithfourdifferentsentences:

  9. UNIFICATION AND LIFTING Thelastunificationfailsbecause x cannottake on thevaluesJohnandElizabeth at thesame time. The problem can be avoidedbystandardizing apart one of thetwosentencesbeingunified, whichmeansrenamingitsvariablestoavoid name clashes. Forexample, we can renamex in to (a newvariable) withoutchangingitsmeaning. Nowtheunificationwillwork:

  10. UNIFICATION AND LIFTING Storage andRetrieval:Wehavetwofunctions STORE and FETCH. STORE(s) stores a sentence s intothe KB and FETCH(q) returnsallunifierssuchthatthequery q unifieswithsomesentence in the KB. The problem usedtoillustrateunification(findingallfactsthatunifywith is an instance of FETCHing. We can make FETCH moreefficientbyensuringthatunificationsareattemptedonlywiththesentencesthathavesomechance of unifying. EXAMPLE:There is nopoint in tryingtounifywith. We can avoidsuchunificationsbyindexingthefacts in KB. A simpleschemecalledpredicateindexingputsalltheKnowsfacts in onebucketandalltheBrotherfacts in another. Thebuckets can be stored in a hashtableforefficientsuccess. EXAMPLE:Supposethatthetaxauthoritieswanttokeeptrack of whoemployswhom, using a predicate.

  11. UNIFICATION AND LIFTING Given a sentenceto be stored, it is possibletoconstructindicesforallpossiblequeriesthatunifywith it. Forthefact, thequeriesare Thesequeries form a subsumptionlatticeas shown in figure:

  12. FORWARD CHAINING First-OrderDefiniteClauses: Definiteclausesare a suitable normal form forusewithGeneralizedModusPonens. Not everyknowledgebase can be convertedinto a set of definiteclauses, but many can. Considerthefollowing problem: Thelawsaysthat it is a crimefor an Americantosellweaponstohostilenations. ThecountryNono, an enemy of America, has somemissiles, andall of itsmissilesweresoldto it byColonel West, who is an American. Wewillprovethat West is a criminal. First, first-orderdefiniteclauses: • ‘’… it is a crimefor an Americantosellweaponstohostilenations’’: • ‘’Nono … has somemissiles.’’ Thesentence is transformedintotwodefiniteclausesbyExistentialElimination, with a newconstant: • ‘’All of itsmissilesweresoldto it byColonel West’’:

  13. FORWARD CHAINING • Wewillalsoneedtoknowthatmissilesareweapons: • andwemustknowthat an enemy of Americacounts as ‘’hostile’’: • ‘’West,who is American ….’’: • ‘’ThecountryNono, an enemy of America ….’’: This KB containsnofunctionsymbolsand is therefore an instance of theclass of DatalogKB which is a set of first-orderdefiniteclauseswithnofunctionsymbols. A Simple ForwardChaining:Theimplicationsentencesare (9.1),(9.4),(9.5) and (9.6). Twoiterationsarerequired: • On thefirstiteration, rule (9.1) has unsatisfiedpremises. Rule(9.4) is satisfiedwith, and is added. Rule(9.5) is satisfiedwith, and is added. Rule(9.6) is satisfiedwith, and is added.

  14. FORWARD CHAINING • On theseconditeration, rule (9.1) is satisfiedwith, and is added. Theprooftree is as shownbelow:

  15. BACKWARD CHAINING Thisalgorithmtakesthefirstgoal in thelistandfindseveryclause in KB whosepositiveliteral, orhead, unifieswiththegoal. Eachsuchclausecreates a newrecursivecall in whichthepremise, orbody, of theclause is addedtothegoalstack. Rememberthatthefactsareclauseswith a head but no body, sowhen a goalunifieswith a knownfact, nonewsubgoalsareaddedtothestackandthegoal is solved. Thefigureshowstheprooftree:

  16. RESOLUTION Conjunctive Normal Form for First-OrderLogic: Toillustratetheprocedurebytranslatingthesentence ‘’Everyonewholovesallanimals is lovedbysomeone’’ Thestepsare as follows: • Eliminateimplications: • Moveinwards: Thesentencenowreads ‘’Eitherthere is someanimalthat x doesn’tlove, orsomeoneloves x’’. Clearly, themeaning of theoriginalsentence is preserved. • Standardize variables: • Skolemize: where F and G areskolemfunctions. • Dropuniversalquantifiers: • Distributeover:

  17. RESOLUTION TheResolutionInferenceRule:First example is thecrimeexample as shown: Thestructure:Beginningwiththegoalclauseand, resolvingagainstclausesfrom KB untiltheemptyclause is generated. CNF SENTENCES

  18. RESOLUTION Second examplemakesuse of Skolemizationandinvolvesclausesthatare not definiteclauses. The problem as follows: Everyonewholovesallanimals is lovedbysomeone. Anyonewhokills an animal is lovedbynoone. Jacklovesallanimals. EitherJackorCuriositykilledthecat, who is named Tuna. DidCuriositykillthecat? First, weexpresstheoriginalsentences, some background knowledge, andthenegatedgoal G in first-orderlogic:

  19. RESOLUTION Now, theconversionprocedure is usedtoconverteachsentenceto CNF:

  20. RESOLUTION TheParaphrasedProof: • SupposeCuriositydid not kill Tuna. • WeknowthateitherJackorCuriositydid; thusJackmusthave. • Now, Tuna is a catandcatsareanimals, so Tuna is an animal. • Becauseanyonewhokills an animal is lovedbynoone, weknowthatnoonelovesJack. • On theotherhand, Jacklovesallanimals, sosomeoneloveshim. • Sowehave a contradiction. Therefore, Curiositykilledthecat. Completeness of Resolution:

  21. RESOLUTION Thebasicstructure of theproof as follows: • First, weobservethatif S is unsatisfiable, thenthereexists a particular set of groundinstancesof theclauses of S suchthatthis set is alsounsatisfiable. • Wethenappealtothegroundresolutiontheorem, whichstatesthatpropositionalresolution is completeforgroundsentences. • Wethenuse a lifting lemmatoshowthat,foranypropositionalresolutionproofusingthe set of groundsentences, there is a correspondingfirst-orderresolutionproofusingthefirst-ordersentencesfromwhichthegroundsentenceswereobtained. Forfirst step, wewillneedthreenewconcepts: • Herbranduniverse: EXAMPLE:If S containsjusttheclause, then is thefollowinginfinite set of groundterms: • Saturation: If S is a set of clausesand P is a set of groundterms, then P(S), thesaturation of S withrespectto P, is the set of allgroundclausesobtainedbyapplyingallpossibleconsistentsubstitutions of groundterms in P withvariables in S.

  22. RESOLUTION • Herbrand Base: If S containssolelytheclausejustgiven, then is theinfinite set of clauses: Herbrand’sTheorem: If a set of clauses is unsatisfiable, thenthereexists a finitesubset of that is alsounsatisfiable. Step 2 is appliedandfor step 3: • Lifting Lemma: Itlifts a proof step fromgroundclausesupto general first-orderclauses. DealingwithEqualtiy:Therearethreedistinctapproaches. • Toaxiomatizeequality:Theequality is reflexive, symmetricandtransitive. We can substituteequalsforequals in anypredicateorfunction. Soweneedthreebasicaxioms, andthenoneforeachpredicateandfunction:

  23. RESOLUTION 2. An additionalinferencerule: • Demodulation: Foranyterms, where and is a literalcontaining z: Demodulation is typicallyusedforsimplifyingexpressionsusingcollections of assertionssuch as andso on. • Paramodulation: For anyterms, where, Unlikedemodulation, paramodulationyields a completeinferenceprocedureforfirst-orderlogicwithequality. 3. An extendedunificationalgorithm:Forexample, theterms 1+2 and 2+1 normallyare not unifiable, but a unificationalgorithmthatknowsthatcouldunifythemwiththeemptysubstitution.

  24. RESOLUTION ResolutionStrategies: • UnitPreference: Thisstrategyprefersto do resolutionswhereone of thesentences is a singleliteral (alsoknown as a unitclause). The idea behindthestrategy is thatwearetryingtoproduce an emptyclause, so it might be a good idea topreferinstancesthatproduceshorterclauses. EXAMPLE:Resolving a unitsentence (such as P) withanyothersentence (such as ) alwaysyield a clause (in thiscase, ) that is shorterthantheotherclause. • Set of Support: Itstartsbyidentifying a subset of thesentencescalledtheset of support.Everyresolutioncombines a sentencefromthe set of supportwithanothersentence, andaddstheresolventintothe set of support. • InputResolution:Here, everyresolutioncombinesone of theinputsentences withsomeothersentence. Shortly, thespace of prooftrees of thisshape is smallerthanthespace of allproofgraphs. *Thelinearresolutionstrategy is a slightgeneralizationthatallowsPandQto be resolvedtogetherifeitherP is in theoriginalKB orifP is an ancestor of Q in theprooftree. 4. Subsumption: Thesubsumptionmethodeliminatesallsentencesthataresubsumedby an existingsentence in the KB. Subsumptionhelpskeepthe KB small, andthushelpskeepthesearchspacesmall.

  25. CONCLUSION • A firstapproachusesinferencerulesforinstantiatingquantifiers in ordertopropositionalizetheinference problem. Typically, thisapproach is veryslow. • Theuse of unificationtoidentifyappropriatesubstitutionsforvariableseliminatestheinstantiation step in first-orderproofs, makingtheprocessmuchmoreefficient. • A liftedversion of ModusPonensusesunificationtoprovide a naturalandpowerfulinferencerule, generalizedModusPonens. Theforwardchainingandbackwardchainingalgorithmsapplythisruletosets of definiteclauses. • GeneralizedModusPonens is completefordefiniteclauses. ForDatalogprogramsconsisting of function-freedefiniteclauses, entailment is decidable. • Forwardchaining is used in deductivedatabases, where it can be combinedwithrelationaldatabaseoperations. It is alsoused in productionsystems, whichperformefficientupdateswithverylargerulesets.

  26. CONCLUSION • Forwardchaining is completeforDatalogprogramsandruns in polynomial time. • Backwardchaining is used in logicprogrammingsystemssuch as Prolog, whichemploysophisticatedcompilertechnologytoprovideveryfastinference. • Backwardchainingsuffersfromredundantinferencesandinfiniteloops; these can be alleviatedbymemorization. • Thegeneralizedresolutioninferenceruleprovides a completeproofsystemforfirst-orderlogic, usingKBs in conjunctive normal form. • Severalstrategiesexistforreducingthesearchspace of a resolutionsystemwithoutcompromisingcompleteness.

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