幾何定理的機(jī)器證明(定稿)

歡迎閱讀本文檔，希望本文檔能對(duì)您有所幫助!歡迎閱讀本文檔，希望本文檔能對(duì)您有所幫助!歡迎閱讀本文檔，希望本文檔能對(duì)您有所幫助!歡迎閱讀本文檔，希望本文檔能對(duì)您有所幫助!歡迎閱讀本文檔，希望本文檔能對(duì)您有所幫助!歡迎閱讀本文檔，希望本文檔能對(duì)您有所幫助!MechanicalTheoremProvinginGeometryGaoJun-yu*,ZhangCheng-dongCangzhouNormalUniversityHebeiprovince,Cangzhoucity,CollegeRoad,061001-mail:gaojunyu8818@126.comAbstractMechanicaltheoremprovingingeometryplaysanimportantroleintheresearchofautomatedreasoning.Inthispaper,weintroducethreekindsofcomputerizedmethodsforgeometricaltheoremproving:thefirstisWu’smethodintheinternationalcommunity,thesecondiseliminationpointmethodandthethirdislowerdimensionmethod.

Keywords：geometrictheorem,Wu’smethod,eliminationpointmethod,lowerdimensionmethod

Copyright?2012UniversitasAhmadDahlan.Allrightsreserved.1.IntroductionMechanicalProving,thatis,mechanizationmethod,istofindamethodwhichcanbecomputedsteeplystepaccordingtoacertainrules.Todayweusuallyreferitas"algorithm".Thealgorithmisappliedinthecomputerprogramming,mathematicalmechanization,andmathematicaltheorems,Thisrealizesmathematicaltheoremtobeprovedwithcomputer.MathematicsinancientChinaisnearlyakindofmechanicmathematics.Today,themethodofCartesiancoordinationgivesthisdirectionasolidstep,andprovidesasimpleandclearmethodfortheproofofgeometric

theorems

mechanization.ThemechanicalthoughtofmathematicsinancientChinamadeadeeplyinfluenceinWuWen-jun'sworkaboutmathematicsmechanization.In1976,academicianWuWen-junbegantoenterthefieldofmathematics

mechanization.Sincethen,heforwardstothemechanizationmethodwhichestablishtheoreticalbasisforthemechanizationofmentalwork.Heprovesalargeclassofelementarygeometryproblemsbycomputer.Itisunprecedentedinourcountry.ThisisamachineprovingmethodandknownasWu’smethodthroughouttheworld.Itisthefirstsystemformechanicprovingmethodanditcangivetheprooffornontrivialtheorem.HemakestheStudyofTheoremProvinginGeometrymoremature[1-6].In1992,AcademicianZhangJing-zhongvisitedtheUSAtoresearchmathematicsmechanizationandcooperatedwithZhouXian-qingandGaoXiaos-han.Theeliminationpointmethodisonethatisbasedonareamethod.ThisbringsreadableProoftoberealizedbycomputerforthefirsttime.Thisresultissignificantforacademicandmechanicaltheoremprovingingeometry.In1998,YangLucreatedlowerdimensionmethod.Heobtainstheachievementinthemechanicproofofinequalities.TheachievementofthismethodisasgoodasWu’smethodandeliminationpointmethod.ItisagreatachievementinthefieldofmechanicaltheoremprovingingeometrybyChinesemathematicians[3-8].Next,wewillintroducethethreemethodsrespectly.2.Wu’smethodWuWen-junpresentsamethodwhichiscalledWu’smethod.ItisbasedonQuaternionoftraditionalmathematics.Thismethodhasbeensolvesaseriesofactualproblemsintheoreticalphysics,computerscienceandotherbasicsciencefields.WecanuseWu’smethodtofindaproofforthegeometrictheoremincomputer.Weintroducethreemainstepsofthismethodasfollows[4-10]:Step1Choosingagoodcoordinatesystem,freevariableandrestrictvariable.Letusdenotethefreevariablesas,andsupposetheyhavenothingtodowiththeconditionsofgeometricproblems.Similarly,letusdenotetherestrictvariablesaswhicharerestrictedbytheconditionsofgeometricproblems.Inthisway,ageometricproblemsturnsintoapolynomialproblem:（1）Theconclusionsofgeometricproblemcanbeexpressedasapolynomialproblem.（2）Or,itcanberepresentedasafamilyofpalynomial.Step2TriangulationAccordingtotherestrictvariable,therearrangeof(1)isreferredastriangulation.Inanotherword,thesystemsofequation(1)ischangedas:（3）Step3GradualDivisionDenotethepolynomialin(3)as,in(2)isdividedby,andtheremainderofdivisionalgorithmisdenotedas.Inordertoavoidthefractionalinquotient,wemultiplyto,thatis,（4）Theremainderisdividedby(5)So,repeatingthisdivision,atlastweget:（6）Then,letusinteratealltheequationsaboveandreplacethecoefficientofofallequationsabovewith,moreover,weobtain:Dividingtwocasestodiscus:withand.Case1:If,then,undertheconditionofandthenon-degeneratecondition,thereis,therequiredresultfollow.Case2:If,thenthepropositionisnottrue.WepresentanexampletoshowthathowtouseWu’sMethodinsolvingproblem.Example1Theproblemis:Themidlineonthehypotenuseinaright-angletriangleequalstohalfofthehypotenuseUsingWu’sMethodtogetthesolutionoftheaboveproblemis:asshowninFigure1,first,choosecoordinatestoright-angletriangle.Thetworight-anglesidesofandareasaxisandaxisrespectively.Thevertexoftherightangleis,.Wetakeasthemidpointofhypotenuseandsetuptheircoordinatesas，，andrespectively.OOB(0,)A(,0)D(,)Figure1Becausearearbitrary,thisindicatethatcoordinatesofarefreevariablesandthemidpointisrestrictedbythehypothesis,sothecoordinatesofarerestrictvariables.ApplyingWu'sMethod,wesolvethisproblemasfollows:Step1Choosingagoodcoordinatesystem,freevariableandrestrictvariable.Supposingisthemidpointofthe,bythemidpointformulawecanget:Usingstep1inWu'sMethod,weonlyneedtoprovethatthe,bythedistantformula:Step2TriangulationBecausejusthastherestrictvariable,andjusthastherestrictvariable,so,themselveshavebeentrianglize.So,itcanbewrittenas:Step3GradualDivisionisdividedby,andthedivisionis:Thatis,（7）where

isdividedby,so,（8）Where.Using(8)toexpress(7),wewillreceiveWhen,Thepropositionhasbeenproved.3.Zhang'sEliminationPointMethodWeintroduceZhang'sEliminationPointMethodasfollows[1-9].ZhangJing-zhonggivesaneffectivemethodforwhatiscalledeliminationpointmethod,themethodisbasedontheancientareamethod,mainlyusedfordeletingtheconstraintpoints.ThisideaofZhang'sEliminationPointMethodisrelativetotheassumedconditionsandareamethod,andtheorderofvanishpointdependsonthefinalconstraintpoints.Thenitiseliminatedfrombacktofrontonebyone.Atlast,theleftpointistotaleliminated,ifthenumberisequaltotherightnumber,thepropositionispermitted.TouseZhang'sEliminationPointMethodeffectively,werepeatthepublicedgetheoreminthefollowing:Next,wegiveacommonlyimportanttheoremofeliminationpointmethod:Publicedgetheorem(1970,Zhang):Ifthelineandlineto,thenPublicedgehavefourcasesareshownas(a)-(d)inFigure2respectively.PPQABM(a)PQABM(b)PABM(c)QPQABM(d)Figure2WegiveanexampletoexpressthePublicedgetheoremofZhang'stheorem.Example2:Theproblemis:Verifythatthediagonalofaparallelogramismutualdivided.Tosolvethisproblemisthefollowing:PutaparallelogramasadiagraminFigure3.1、Doaparallelogram.2、Connectingthediagonalswhichintersectat.Weonlyneedtoverify,.istherestrictionpointwhichfinallymade.So,firstlytoremovethepoint.(Publicedgetheorem)AABCDEFigure3Therefore,wecanprove,andastheabovesimilarway..4.LowerDimensionMethodWiththeestablishmentofWu’smethodandeliminationpointmethod,theMachineProvingofautomatedtheoremprovingofequationtheoremhasbeensolved,buttheMachineProvingofautomatedtheoremproofofequationtheoremhasbeendifficulttoachieve.Therefore,YangLuandmanyotherscholarsworkedfortheestablishmentofanewalgorithmwhichwecalledlowerdimensionmethod.TheworkinthefieldofmachinetheoremprovingbyChinesemathematiciansisamilestone[2-12].Thelowerdimensionmethodcanbedividedintothreecourses:(1)Workoutaboutboundarysurfaceofinequality、、、.(2)Usingtheboundarysurfaceofthefirststeptheparameterspacewasdividedintofinitecelldecompositions,wegetmanyconnectedopensets:、、、,thenfromtheconnectedopensetsweselectcheckpointforatleastoneabbrevd.(3)usingthefinitenumberofcheckpoints、、、toverifythecorrectnessofinequality.Ifeveryestablishedthepropositionistrue,otherwise,thepropositionisfalse.5.ConclusionThethreemethodspresentdifferentwaystodealwithdifferentproblems.Allofthemareimportantinautomatedreasoningfields,whensomeonediscussesaprobleminautomatedreasoningfields,thefirsthe(orshe)wouldconsiderwhichofthethreemethodsisjustbestfortheproblem,andthen,he(orshe)willobtainthebestconsequences.Wehopeourintroductionisgoodforhim(orher),nowandinthefuture.References[1]WuWen-jun.Elementarygeometrytrussproofandmechanization[J].Chinesescience,1997,(6),(inchinese).[2]WuWen-jun.Geometrictheoremmachinethebasicprincipleofproof[M].Beijing:sciencepress,1984,(inchinese).[3]ZhangJing-zhong.Awayoncollocationmethod[J].Mathematicsteacher,1995,(1),(inchinese).[4]ZhangJing-zhong.Thecomputerhowtoworkoutgeometricproblem[M].Beijing:tsi