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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

Proving on Inequalities

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

;onInequalities

;YANGLu(杨路)

;ChengduInstituteofComputerApplications

;E-mail:luyang@guangztc.edu.ca

;fundamentallyonrealalgebraandrealgeometry,andthecomputationalcomplexity

;increasesveryquicklywiththedimension,thatis,thenumberofparameters.Some

;wellknownalgorithmsarecompletetheoreticallybutine~cientinpractice.which

galgorithm

;Baseduponthisalgorithm.agenericprogramcalled”BOTTEMA’’Wasimplemented

;onapersonalcomputer.Morethan1000algebraicandgeometricinequalitiesinclud

;inghundredsofopenproblemshavebeenverifiedinthisway.Thismakesitpossible

roblem.

;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

;,,,

;9)

;9

;4驺一8?一坞一

;

;No.5AutomatedTheoremProvingonInequalities435

;sothencheck/(x,Y)

;sothencheck/(x,Y)

;4633573525

;1073741824

;9267147051

;2147483648

;true

;true

t

;thatis

;9267147051.2316786763

;2147483648,”,536870912’

;fmi=4.315351625

;whereallthe10digitswrittendownaresignificant.Thisisaccurateenoughforgeneral

;purpose.Byaprogramnamed”BOTTEMA”,wefinishedthejobonaPentium

/350using

;Thedichotomoussearchshouldbemuchmoreefficientifwecanfindafinitesetwhich

;theoptimalvaluebelongsto.Thisisdefinitelypossibleandshallbeinvestigatedlater.

;Itwouldbeconcludedfromtheaboveexamplethatthespeedinautomatedtheorem

;provingisalsoofimportance.?

havereasontobelievethatcomputerwil1playamuch

;moreimportantroleinreasoningsciencesinthenextcentury.Peoplewillbeabletoprove

;MethodforElementaryAlgebraandGeometry,publishedinearly1950’s,thealgebraicap

ki’sdecision

;algorithmhasonlygottheoreticalsignificance.thatcouldnotbeusedtoverifyanynon

;trivialalgebraicorgeometricpropositionsinpractice,becauseofitsveryhighcomputational

;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

;approacht’toautomatedtheoremproving.

;Inthepast20years.someemcientprovershave

;beendevelopedbasedondifierentprinciplessuchasGrSbnerBasis[0,1,ParallelNumerical

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

;The

;achievementmakesthestudiesinautomatedprovingenteranewstagethattheproofs

;createdbymachinescancomparewiththosebyhumanbeing,

;whilethedecisionproblem

;andCAI.

;Thepackagesmentionedabovearemainlyvalidtoequalitytypetheorempr

oving,how

;reasoningformanyyears.Theconcerningalgorithmsdependonrealalgebraandrealge..

;ometry,andthecomputationalcomplexityincreasesveryquicklywiththedimension.i.e.

;thenumberofparameters.Somewellknownalgorithmsarecompletetheor

eticallybutinef-

;ficientinpractice,whichcannotverifynontrivialpropositionsinbatches.R

ecentlyChou.

u’smethodwith

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

;TheCDSwouldbeabletosolveavarietyofproblemsinscience,technologyanden

;theoremsofhigherdimensionsorwithmoreparameters.Whenthehypothesescontain

;somealgebraicequations,onemayconsidertoeliminatesomevariablestomakethedl

;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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