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Recent Advances in Automated Theorem Proving on Inequalities

By Brent Hayes,2014-09-22 16:04
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Recent Advances in Automated Theorem Proving on Inequalitiesin,on

    Recent Advances in Automated Theorem

    Proving on Inequalities

Vo1.14NO.5J.Comput.Sci.&Techno1.Sept.1999

    ;RecentAdvancesinAutomatedTheoremProving

    ;onInequalities

    ;YANGLu(杨路)

    ;ChengduInstituteofComputerApplications

    ;ChineseAcademyofSciences,Chengdu61004j,P.R.China

    ;E-mail:luyang@guangztc.edu.ca

    ;AbstractAutomatedtheoremprovingoninequalitiesiSalwaysconsideredas ;adi~culttopicintheareaofautomatedreasoning.Therelevantalgorithmsdepend

    ;fundamentallyonrealalgebraandrealgeometry,andthecomputationalcomplexity

    ;increasesveryquicklywiththedimension,thatis,thenumberofparameters.Some

    ;wellknownalgorithmsarecompletetheoreticallybutine~cientinpractice.which

    ;cannotverifynontrivialpropositionsinbatches.Adimensiondecreasin

    galgorithm

    ;presentedherecantreatradicalseflicientlyandmakethedimensionsthelowest.

    ;Baseduponthisalgorithm.agenericprogramcalled”BOTTEMA’’Wasimplemented

    ;onapersonalcomputer.Morethan1000algebraicandgeometricinequalitiesinclud

    ;inghundredsofopenproblemshavebeenverifiedinthisway.Thismakesitpossible

    optimizationp;tocheckafinitemanyinequalitiesinsteadofsolvingaglobal

    roblem.

    ;Keywordsinequalitywithradicals,rationalization,dichotomoussearch ;1Introduction

    ;LetUSstartfromthefollowingproblem:[]findthesmallestvalueof ;f(x,)=,/+1/x+xy+i/y.

    ;subjectto>0.Y>0.

    ;Ifthereisapower~ltoolforinequalityprovinginelementaryalgebra,onemayapplya

    ;naivealgorithmtotheproblembynomeansotherthanthistoolwhichcandecideagiven

    ;inequalitytobetrueorfalse.Atfirstitiseasytoseethat1<fmin<5.Wethenusea

    ;dichotomoussearchtofindfminapproximately.

;Checktheinequalityf(x,Y)

    ;SOthencheckf(x,Y)?

    ;SOthencheck ;SOthencheck ;SOthencheck ;SOthencheck ;SOthencheck ;SOthencheck ;f(x,Y)

    ;f(x,Y)?

    ;f(x,Y)

    ;f(x,Y)

    ;f(x,Y)

    ;true,

    ;true,

    ;false,

    ;true.

    ;false,

    ;true,

    ;false.

    ;,

    ;false

;ResearchsupportedinpartbyNational’973’ProjectofChinaand’95’KeyPro

    jectonFundamental

    ;ResearchofAcademiaSinica. ;.Ilt/卜营X:

    ;,,,

    ;9)

    ;9

    ;4驺一8?一坞一

    ;

    ;No.5AutomatedTheoremProvingonInequalities435

    ;sothencheck/(x,Y)

    ;sothencheck/(x,Y)

    ;4633573525

    ;1073741824

    ;9267147051

    ;2147483648

    ;true

    ;true

    ;so,afterweproved/disproved33inequalitiesofthesametype,itwasknowntha

    t

    ;thatis

    ;9267147051.2316786763

;2147483648,”,536870912’

    ;fmi=4.315351625

    ;whereallthe10digitswrittendownaresignificant.Thisisaccurateenoughforgeneral

    ;purpose.Byaprogramnamed”BOTTEMA”,wefinishedthejobonaPentium

    /350using

    ;timeabout100seconds.

    ;Thedichotomoussearchshouldbemuchmoreefficientifwecanfindafinitesetwhich

    ;theoptimalvaluebelongsto.Thisisdefinitelypossibleandshallbeinvestigatedlater.

    ;Itwouldbeconcludedfromtheaboveexamplethatthespeedinautomatedtheorem

    ;provingisalsoofimportance.?

    havereasontobelievethatcomputerwil1playamuch

    ;moreimportantroleinreasoningsciencesinthenextcentury.Peoplewillbeabletoprove

    ;theoremsclassbyclassinsteadofonebyone.SinceTarski’s_2.well—known

    work.ADecision

    ;MethodforElementaryAlgebraandGeometry,publishedinearly1950’s,thealgebraicap

    ;proacheshavemaderemarkableprogressinautomatedtheoremproving.Tars

ki’sdecision

    ;algorithmhasonlygottheoreticalsignificance.thatcouldnotbeusedtoverifyanynon

    ;trivialalgebraicorgeometricpropositionsinpractice,becauseofitsveryhighcomputational

    ;complexity.SomesubstantialimprovementsweremadebySeidenberg[a.

    ;Collins[4andoth

    ;ersafterwards,butitwasstillfarawayfrommechanicallyprovingnontrivia

    ltheorems

    ;batchbybatch,evenclassbyclass.Thesituationdidnotchangeuntil?

    Wentsfin[5,6

    ;proposedin1977anewdecisionprocedureforprovinggeometrytheoremsof”equalitytype”,

    ;i.e.thehypothesesandconclusionsofthestatementsconsistofpolynomialequationsonly.

    ;Thisisaveryemcientmethodformechanicallyprovingelementarygeometrytheoremsrof

    ;equalitytype1.S.C.Chout1hassuccessfullyimplementedWu’smethodfor512examples

    ;whichincludealmostallthewellknownorhistoricallyinterestingtheorem

    sinelementary

    ;geometry,anditwasreportedthatformostoftheexamplestheCPUtimespent

wasonly

    ;fewsecondseach.orlessthan1second!

    ;Thesuccessof

    ?’smethodhasinspiredintheworldtheadvancesofthealgebraic

    ;approacht’toautomatedtheoremproving.

    ;Inthepast20years.someemcientprovershave

    ;beendevelopedbasedondifierentprinciplessuchasGrSbnerBasis[0,1,ParallelNumerical

    ;Method[,andsoon.Especially.J.Z.Zhangandhiscolleaguesgavethealgorithmsand

    ;programsforautomaticallyproducingreadableproofsofgeometrytheorems[.

    ;The

    ;achievementmakesthestudiesinautomatedprovingenteranewstagethattheproofs

    ;createdbymachinescancomparewiththosebyhumanbeing,

    ;whilethedecisionproblem

    ;wasplayingaleadingrole.Ithasalsoimportantapplicationstomathematicsmechanization

    ;andCAI.

    ;Thepackagesmentionedabovearemainlyvalidtoequalitytypetheorempr

    oving,how

    ;ever,automatedinequalityprovinghasbeenadimculttopicintheareaofautomated

    ;reasoningformanyyears.Theconcerningalgorithmsdependonrealalgebraandrealge..

    ;ometry,andthecomputationalcomplexityincreasesveryquicklywiththedimension.i.e.

    ;thenumberofparameters.Somewellknownalgorithmsarecompletetheor

    eticallybutinef-

    ;ficientinpractice,whichcannotverifynontrivialpropositionsinbatches.R

    ecentlyChou.

    ;Gaoeta1.[18,191madehelpfulapproachesinthisaspectbycombining?

    u’smethodwith

    ;CAD(CylindricalAlgebraicDecomposition)algorithmorothers.??

    ntsfin[20,21]pro-

    ;posedthefinitekernelprinciple,combiningwithhiselimination, ;whereofhemakeuseto

    ;

    ;436J.Comput.Sci.&TechnolVo1.14

    ;proveinequalitiesandsolveoptimizationproblems.L.Yangandhiscolleagues[Jintro

    ;ducedastrongtool,acompletediscriminationsystem(CDS)ofpolynomials,forinequality

;reasoning.BymeansofCDSagenericprogramcalled”DISCOVERER”was

    alsoimple

    ;mentedonPCcomputersthatisabletodiscovernewinequalities,withoutreqmrmgus

    ;toputforwardanyconjecturesbeforehand.Forexample,bymeansofthisprogram,we

    ;haverediscovered37inequalitiesinthefirstchapterofthefamousmonograph[25J,”R.ecent

    ;AdvancesinGeometricInequalities”,andfoundthreemistakesthere.

    ;TheCDSwouldbeabletosolveavarietyofproblemsinscience,technologyanden

    ;gineeringthataskforrealsolutions,butitwouldbeinvalidforautomaticallyprovingthe

    ;theoremsofhigherdimensionsorwithmoreparameters.Whenthehypothesescontain

    ;somealgebraicequations,onemayconsidertoeliminatesomevariablestomakethedl

    ;mensionlower.Inthisway.however.usuallywehavetodealwithparametricradicals.A

    ;dimensiondecreasingalgorithmintroducedinf26,27]cantreatradicalsefficientlyandmake

    ;thedimensionsthelowest.Basedonthisalgorithm,agenericprogramcalled”

BOTTEMA”

    ;wasimplementedonaPCcomputer.Morethan1000algebraicandgeometricinequalities

    ;includinghundredsopenproblemshavebeenverifiedinthisway.ThetotalCPUtimespent

    ;forproving120basicinequalitiesfromBottema’s28

    monograph,”GeometricInequalities”

    ;onaPentium/200,was20oddsecondsonly.

    ;2AnIllustrationtoDimensionDecreasingAlgorithm

    ;Forpopularity,weshowthemainpointofouralgorithmwiththefollowinginequalitytype

    ;proposition.

    ;Proposition1.Givenrealnumbersu,u,w,,Y,zsatisfyingthefollowing9conditions,

    ;u2+6u

    ;v2+6yv.

    ;w2J-6zw

    ;Y2+2yz

    ;z2+2zx

    ;2+2xy

    ;4xy

    ;4yz

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