5 Normal Forms and Skolemization Traditional Study of normal forms motivated by reduction of logical concepts e64259cient data structures for theorem proving The main problem in 64257rstorder logic is the t ID: 25216
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3.5NormalFormsandSkolemization(Traditional)Studyofnormalformsmotivatedbyreductionoflogicalconcepts,ecientdatastructuresfortheoremproving.Themainproblemin rst-orderlogicisthetreatmentofquanti ers.Thesubsequentnormalformtransformationsareintendedtoeliminatemanyofthem.PrenexNormalFormPrenexformulashavetheform:::QF;whereisquanti er-freeand2f89g;wecall:::Qthequanti erpre xandthematrixoftheformula.ComputingprenexnormalformbytherewriterelationQxF Qx((QxFGQyyy=xG;yfresh;2f^_g((QxF Qyyy=x;yfreshFQxG))QyFGGy=x]);yfresh;2f^!gHere denotesthequanti erdualto,i.e., and SkolemizationIntuition:replacementofbyaconcretechoicefunctioncomputingfromalltheargumentsdependson.Transformation(tobeappliedoutermost,notinsubformulas):;:::;xyF;:::;xxf(x1;:::;x=ywhere,wherearity)=,isanewfunctionsymbol(Skolemfunction).Together:|{z}prenex|{z}prenex,noTheorem3.9Let,andasde nedaboveandclosed.Then48 (i)andareequivalent.(ii)buttheconverseisnottrueingeneral.(iii)satis able(w.r.t.-Algsatis able(w.r.t.-Alg)where=(\n\nSKF;),if=(\n)ClausalNormalForm(ConjunctiveNormalForm)^:_:::^�^?_�_?TheserulesaretobeappliedmoduloassociativityandcommutativityofandThe rst verules,plustherule(),computethenegationnormalform(NNF)ofaformula.TheCompletePicture:::Qquanti er-free);:::;xquanti er-free);:::;x {z leaveout=1=1ij {z clauses {z ;:::;Ciscalledtheclausal(normal)form(CNF)ofNote:thevariablesintheclausesareimplicitlyuniversallyquanti ed.Theorem3.10Letbeclosed.Then.(Theconverseisnottrueingeneral.)Theorem3.11Letbeclosed.Thenissatis ablei issatis ablei issatis able49 OptimizationHereislotsofroomforoptimizationsinceweonlycanpreservesatis abilityanyway:sizeoftheCNFexponentialwhendonenaively;butseethetransformationsweintroducedforpropositionallogicwanttopreservetheoriginalformulastructure;wantsmallarityofSkolemfunctions(follows)3.6GettingsmallSkolemFunctionsproduceanegationnormalform(NNF)applyminiscopingrenameallvariablesskolemizeNegationNormalForm(NNF)ApplytherewriterelationNNFistheoverallformula:NNFifF=pandF=phaspositivepolarityNNF^:ifF=pandF=phasnegativepolarityQxGNNF QxNNF^:NNF_:NNF::NNFMiniscopingApplytherewriterelationMS.Forthebelowrulesweassumethatoccursfreelyin,butdoesnotoccurfreelyinQxMSQxGQxMSQxGMSxG^8xHMSxG_9xH50 VariableRenamingRenameallvariablesinsuchthattherearenotwodi erentpositionsp;qwithF=pQxGandF=qQxHStandardSkolemizationLetbetheoverallformula,thenapplytherewriterule:xHSKKf(y1;:::;y=xifF=pxHandhasminimallength,;:::;yarethefreevariablesinxHisanewfunctionsymbol,arity)=3.7HerbrandInterpretationsFromnowanweshallconsiderPLwithoutequality.\nshallcontainsatleastoneconstantsymbol.Herbrandinterpretation(over)isa-algebrasuchthat=T(=thesetofgroundtermsover):(;:::;s;:::;s;farity)= ;:::;)=:::Inotherwords,valuesare xedtobegroundtermsandfunctionsare xedtobethetermconstructors.Onlypredicatesymbols,arity)=maybefreelyinterpretedasrelationsProposition3.12EverysetofgroundatomsuniquelydeterminesaHerbrandinter-pretationvia;:::;s;:::;s51