Outline KSubramani11LaneDepartmentofComputerScienceandElectricalEngineeringWestVirginiaUniversityCompletenessCompactnessandInexpressibility Subramani FirstOrderLogic Outline Outline 1 Completenesso ID: 509706
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Outline AnIntroductiontoFirst-OrderLogic K.Subramani11LaneDepartmentofComputerScienceandElectricalEngineeringWestVirginiaUniversityCompleteness,CompactnessandInexpressibility Subramani First-OrderLogic Outline Outline 1 CompletenessofproofsystemforFirst-OrderLogic ThenotionofCompleteness TheCompletenessProof 2 ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Subramani First-OrderLogic Outline Outline 1 CompletenessofproofsystemforFirst-OrderLogic ThenotionofCompleteness TheCompletenessProof 2 ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ThenotionofCompleteness TheCompletenessProof Outline 1 CompletenessofproofsystemforFirst-OrderLogic ThenotionofCompleteness TheCompletenessProof 2 ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ThenotionofCompleteness TheCompletenessProof SoundnessandCompleteness Theorem Soundness:If` ,thenj= . Theorem Completeness(G¨odel'straditionalform):Ifj= ,then` . Theorem Completeness(G¨odel'saltenateform):Ifisconsistent,thenithasamodel. Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ThenotionofCompleteness TheCompletenessProof SoundnessandCompleteness(contd.) Theorem Thetraditionalcompletenesstheoremfollowsfromthealternateformofthecompletenesstheorem. Proof. Assumethatj= . ItfollowsthatanymodelMthatsatisesalltheexpressionsin,alsosatises andhencefalsies: . Thus,theredoesnotexistamodelthatsatisesalltheexpressionsin[f: g.Itfollowsthat[f: gisinconsistent.ButusingtheContradictiontheorem,itfollowsthat` . Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ThenotionofCompleteness TheCompletenessProof SoundnessandCompleteness(contd.) Theorem Thetraditionalcompletenesstheoremfollowsfromthealternateformofthecompletenesstheorem. Proof. Assumethatj= . ItfollowsthatanymodelMthatsatisesalltheexpressionsin,alsosatises andhencefalsies: .Thus,theredoesnotexistamodelthatsatisesalltheexpressionsin[f: g. Itfollowsthat[f: gisinconsistent.ButusingtheContradictiontheorem,itfollowsthat` . Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ThenotionofCompleteness TheCompletenessProof SoundnessandCompleteness(contd.) Theorem Thetraditionalcompletenesstheoremfollowsfromthealternateformofthecompletenesstheorem. Proof. Assumethatj= .ItfollowsthatanymodelMthatsatisesalltheexpressionsin,alsosatises andhencefalsies: . Thus,theredoesnotexistamodelthatsatisesalltheexpressionsin[f: g.Itfollowsthat[f: gisinconsistent. ButusingtheContradictiontheorem,itfollowsthat` . Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ThenotionofCompleteness TheCompletenessProof SoundnessandCompleteness(contd.) Theorem Thetraditionalcompletenesstheoremfollowsfromthealternateformofthecompletenesstheorem. Proof. Assumethatj= .ItfollowsthatanymodelMthatsatisesalltheexpressionsin,alsosatises andhencefalsies: .Thus,theredoesnotexistamodelthatsatisesalltheexpressionsin[f: g. Itfollowsthat[f: gisinconsistent.ButusingtheContradictiontheorem,itfollowsthat` . Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ThenotionofCompleteness TheCompletenessProof SoundnessandCompleteness(contd.) Theorem Thetraditionalcompletenesstheoremfollowsfromthealternateformofthecompletenesstheorem. Proof. Assumethatj= .ItfollowsthatanymodelMthatsatisesalltheexpressionsin,alsosatises andhencefalsies: .Thus,theredoesnotexistamodelthatsatisesalltheexpressionsin[f: g.Itfollowsthat[f: gisinconsistent. ButusingtheContradictiontheorem,itfollowsthat` . Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ThenotionofCompleteness TheCompletenessProof SoundnessandCompleteness(contd.) Theorem Thetraditionalcompletenesstheoremfollowsfromthealternateformofthecompletenesstheorem. Proof. Assumethatj= .ItfollowsthatanymodelMthatsatisesalltheexpressionsin,alsosatises andhencefalsies: .Thus,theredoesnotexistamodelthatsatisesalltheexpressionsin[f: g.Itfollowsthat[f: gisinconsistent.ButusingtheContradictiontheorem,itfollowsthat` . Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ThenotionofCompleteness TheCompletenessProof Outline 1 CompletenessofproofsystemforFirst-OrderLogic ThenotionofCompleteness TheCompletenessProof 2 ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ThenotionofCompleteness TheCompletenessProof ProofSketchofCompletenessTheorem Proof. http://www.maths.bris.ac.uk/rp3959/firstordcomp.pdf Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidity CompactnessModelCardinalityL¨owenheim-SkolemTheoremInexpressibilityofReachability Outline 1 CompletenessofproofsystemforFirst-OrderLogic ThenotionofCompleteness TheCompletenessProof 2 ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidity CompactnessModelCardinalityL¨owenheim-SkolemTheoremInexpressibilityofReachability Validity Theorem VALIDITYisRecursivelyenumerable. Proof. Followsinstantaneouslyfromthecompletenesstheorem. Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinalityL¨owenheim-SkolemTheoremInexpressibilityofReachability Outline 1 CompletenessofproofsystemforFirst-OrderLogic ThenotionofCompleteness TheCompletenessProof 2 ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinalityL¨owenheim-SkolemTheoremInexpressibilityofReachability Compactness Theorem Ifallnitesubsetsofasetofsentencesaresatisable,thensois. Proof. Assumethatisunsatisable,butallnitesubsetsofaresatisable. Asperthecompletenesstheorem,thereisaproofofacontradictionfrom,say` ^: .However,thisproofhasnitelength!Therefore,itcaninvolveonlyanitesubsetof! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinalityL¨owenheim-SkolemTheoremInexpressibilityofReachability Compactness Theorem Ifallnitesubsetsofasetofsentencesaresatisable,thensois. Proof. Assumethatisunsatisable,butallnitesubsetsofaresatisable.Asperthecompletenesstheorem,thereisaproofofacontradictionfrom,say` ^: . However,thisproofhasnitelength!Therefore,itcaninvolveonlyanitesubsetof! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinalityL¨owenheim-SkolemTheoremInexpressibilityofReachability Compactness Theorem Ifallnitesubsetsofasetofsentencesaresatisable,thensois. Proof. Assumethatisunsatisable,butallnitesubsetsofaresatisable.Asperthecompletenesstheorem,thereisaproofofacontradictionfrom,say` ^: .However,thisproofhasnitelength!Therefore,itcaninvolveonlyanitesubsetof! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactness ModelCardinality L¨owenheim-SkolemTheoremInexpressibilityofReachability Outline 1 CompletenessofproofsystemforFirst-OrderLogic ThenotionofCompleteness TheCompletenessProof 2 ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactness ModelCardinality L¨owenheim-SkolemTheoremInexpressibilityofReachability ModelSize Theorem Ifasentencehasamodel,ithasacountablemodel. Proof. ThemodelMconstructedintheproofofthecompletenesstheoremiscountable,sincethecorrespondingvocabularyiscountable. Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Outline 1 CompletenessofproofsystemforFirst-OrderLogic ThenotionofCompleteness TheCompletenessProof 2 ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Query Doallsentenceshaveinnitemodels? Theorem Ifasentence hasnitemodelsofarbitrarylargecardinality,thenithasaninnitemodel. Proof. Considerthesentence k=9x19x2:::9xk^1ijk:(xi=xj). kcannotbesatisedwithamodelhavinglessthankelements. Assumethat hasarbitrarilylargemodels,butnoinnitemodels.Let= [f k:k=2;3;:::g.IfhasamodelM,Mcanneitherbenitenorinnite.Thus,doesnothaveamodel. .Bythecompactnesstheorem,anitesubsetDdoesnothaveamodel. mustbeinD.Letkdenotethelargestinteger,suchthat k2D.Butthereisalargeenoughmodelthatsatisesboth (hypothesis)and k! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Query Doallsentenceshaveinnitemodels? Theorem Ifasentence hasnitemodelsofarbitrarylargecardinality,thenithasaninnitemodel. Proof. Considerthesentence k=9x19x2:::9xk^1ijk:(xi=xj). kcannotbesatisedwithamodelhavinglessthankelements.Assumethat hasarbitrarilylargemodels,butnoinnitemodels.Let= [f k:k=2;3;:::g.IfhasamodelM,Mcanneitherbenitenorinnite. Thus,doesnothaveamodel..Bythecompactnesstheorem,anitesubsetDdoesnothaveamodel. mustbeinD.Letkdenotethelargestinteger,suchthat k2D.Butthereisalargeenoughmodelthatsatisesboth (hypothesis)and k! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Query Doallsentenceshaveinnitemodels? Theorem Ifasentence hasnitemodelsofarbitrarylargecardinality,thenithasaninnitemodel. Proof. Considerthesentence k=9x19x2:::9xk^1ijk:(xi=xj). kcannotbesatisedwithamodelhavinglessthankelements.Assumethat hasarbitrarilylargemodels,butnoinnitemodels.Let= [f k:k=2;3;:::g.IfhasamodelM,Mcanneitherbenitenorinnite.Thus,doesnothaveamodel. .Bythecompactnesstheorem,anitesubsetDdoesnothaveamodel. mustbeinD.Letkdenotethelargestinteger,suchthat k2D.Butthereisalargeenoughmodelthatsatisesboth (hypothesis)and k! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Query Doallsentenceshaveinnitemodels? Theorem Ifasentence hasnitemodelsofarbitrarylargecardinality,thenithasaninnitemodel. Proof. Considerthesentence k=9x19x2:::9xk^1ijk:(xi=xj). kcannotbesatisedwithamodelhavinglessthankelements.Assumethat hasarbitrarilylargemodels,butnoinnitemodels.Let= [f k:k=2;3;:::g.IfhasamodelM,Mcanneitherbenitenorinnite.Thus,doesnothaveamodel..Bythecompactnesstheorem,anitesubsetDdoesnothaveamodel. mustbeinD.Letkdenotethelargestinteger,suchthat k2D.Butthereisalargeenoughmodelthatsatisesboth (hypothesis)and k! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Query Doallsentenceshaveinnitemodels? Theorem Ifasentence hasnitemodelsofarbitrarylargecardinality,thenithasaninnitemodel. Proof. Considerthesentence k=9x19x2:::9xk^1ijk:(xi=xj). kcannotbesatisedwithamodelhavinglessthankelements.Assumethat hasarbitrarilylargemodels,butnoinnitemodels.Let= [f k:k=2;3;:::g.IfhasamodelM,Mcanneitherbenitenorinnite.Thus,doesnothaveamodel..Bythecompactnesstheorem,anitesubsetDdoesnothaveamodel. mustbeinD.Letkdenotethelargestinteger,suchthat k2D.Butthereisalargeenoughmodelthatsatisesboth (hypothesis)and k! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinalityL¨owenheim-SkolemTheorem InexpressibilityofReachability Outline 1 CompletenessofproofsystemforFirst-OrderLogic ThenotionofCompleteness TheCompletenessProof 2 ConsequencesoftheCompletenesstheorem ComplexityofValidity Compactness ModelCardinality L¨owenheim-SkolemTheorem InexpressibilityofReachability Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinalityL¨owenheim-SkolemTheorem InexpressibilityofReachability REACHABILITY REACHABILITY GivenadirectedgraphGandtwonodesxandyinG,isthereadirectedpathfromxtoyinG? Theorem Thereisnorst-orderexpression withtwofreevariablesxandy,suchthat -GraphsexpressesREACHABILITY. Proof. Assumethatthereexistssucha .Considerthesentence, 0= 0^ 1^ 2,where, 0=(8x)(8y) 1=(8x)(9y)G(x;y)^(8x)(8y)(8z)((G(x;y)^G(x;z))!(y=z)) 2=(8x)(9y)G(y;x)^(8x)(8y)(8z)((G(y;x)^G(z;x))!(y=z)) Arbitrarilylargemodelsarepossiblefor 0,butnoinnitemodels! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinalityL¨owenheim-SkolemTheorem InexpressibilityofReachability REACHABILITY REACHABILITY GivenadirectedgraphGandtwonodesxandyinG,isthereadirectedpathfromxtoyinG? Theorem Thereisnorst-orderexpression withtwofreevariablesxandy,suchthat -GraphsexpressesREACHABILITY. Proof. Assumethatthereexistssucha .Considerthesentence, 0= 0^ 1^ 2,where, 0=(8x)(8y) 1=(8x)(9y)G(x;y)^(8x)(8y)(8z)((G(x;y)^G(x;z))!(y=z)) 2=(8x)(9y)G(y;x)^(8x)(8y)(8z)((G(y;x)^G(z;x))!(y=z))Arbitrarilylargemodelsarepossiblefor 0,butnoinnitemodels! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinalityL¨owenheim-SkolemTheorem InexpressibilityofReachability REACHABILITY REACHABILITY GivenadirectedgraphGandtwonodesxandyinG,isthereadirectedpathfromxtoyinG? Theorem Thereisnorst-orderexpression withtwofreevariablesxandy,suchthat -GraphsexpressesREACHABILITY. Proof. Assumethatthereexistssucha .Considerthesentence, 0= 0^ 1^ 2,where, 0=(8x)(8y) 1=(8x)(9y)G(x;y)^(8x)(8y)(8z)((G(x;y)^G(x;z))!(y=z)) 2=(8x)(9y)G(y;x)^(8x)(8y)(8z)((G(y;x)^G(z;x))!(y=z)) Arbitrarilylargemodelsarepossiblefor 0,butnoinnitemodels! Subramani First-OrderLogic Completeness ConsequencesoftheCompletenesstheorem ComplexityofValidityCompactnessModelCardinalityL¨owenheim-SkolemTheorem InexpressibilityofReachability REACHABILITY REACHABILITY GivenadirectedgraphGandtwonodesxandyinG,isthereadirectedpathfromxtoyinG? Theorem Thereisnorst-orderexpression withtwofreevariablesxandy,suchthat -GraphsexpressesREACHABILITY. Proof. Assumethatthereexistssucha .Considerthesentence, 0= 0^ 1^ 2,where, 0=(8x)(8y) 1=(8x)(9y)G(x;y)^(8x)(8y)(8z)((G(x;y)^G(x;z))!(y=z)) 2=(8x)(9y)G(y;x)^(8x)(8y)(8z)((G(y;x)^G(z;x))!(y=z))Arbitrarilylargemodelsarepossiblefor 0,butnoinnitemodels! Subramani First-OrderLogic