MULTIPLEREPRESENTATIONS OF KNOWLEDGEIN A MECHANICS PROBLEM~SOLVER

JohandeKleer

Artificial IntelligenceLaboratoryMassachusettsInstitute of Technology

Cambridge,Massachusetts02139

Abstract

Expert problem-solving programshave focused onworking problems which humansconsiderdifficult. Oddly.many such problem-solverscould not solve less difficultversionsof the problemsaddressedby their expertise. Thisshortcomingalsocontributedto theseprograms’inability tosolve harderproblems. To overcomethis ‘paradox’ requiresmultiple representationsof knowledge,inferencingschemesforeach,andcommunicationschemesbetweenthem.

This paperpresentsa program,NEWTON, applyingthis idea to the domain of simple classical mechanics.NEWTON employs the method of envisionment,wherebysimplequestionsmay be answereddirectly, and plansproducedfor solving more complex problems. Envisioning enablesNEWTON to usequalitativeargumentswhen possible,withresortsto mathematicalequationsonly if thequalitativereasoningfalls to produceasolution. -

Introduction

Expert problem-solvingprogramshave focused onworking problemswhich humansconsiderdifficult. Charniakdemonstratedthe expertiseof CARPS[68) by using it to solveproblemsfrom afreshmancalculustext. The achievementsofMACSYMA tMathlab 74) are touted by exhibiting soniccomplicatedexpressionsit can integrate. Thereis no doubtthat thesetwo problem areasaredifficult, but before weattributeexpertiseto theseprogramswe should first examinethedepthof their understandingof the problem areas. Iproposeonecriterion: As well as beingable to solve th~difficultproblems,the expertproblem-solvershouldbe able to solve simplerversionsof a problemwit/i qualitatively simpler techniques. Wewill seethat the inability to solve simpleproblems is the sourceof many of the difficulties problem-solversencounter inattackingharderproblems.

My experienceteachingelectrical network theorysuggeststhat this criterion also applies to students. Considerthe problemof determiningthe averagepower-dissipatedby anetwork. A general techniquewhich solves this problem is towrite down Kirchoff’s current law for all the nodesand solvethe resultingequations. A poor student alwaysapplies thistechniqueimmediately,while the betterstudentlooks first to seewhethersomesimplificationsarepossible. For example,if thenetwork consistedof only capacitorsand inductors, the poorstudent still sets up equations while the better studentimmediately replies zero since capacitolsand inductors do notdissipatepower. The reasonthe student who immediately setsup equationsdoesworse, is that without first simplif ying theproblemthe resultingequationsareoften unmanageable.If thetopology of the network wasunspecified,the poor studentwould be unableto even set up the equationswhile the betterstudentwould still be able to solve the problem.

To determinewhetheran the object releasedat AreachesB on the roughenedtrack requiresa carefulanalysisofthe shapeand frictional propertiesof the track.The slightly different problem where friction is zero can besolvedwith the sametechnique. This would he a ratherstupidsincea simple comparisonof the relative heightsof A and B

Knowledge Repr299

would solve the problem directly. The mark of an expert isthat such qualitatively simpler problems are attacked byqualitatively dii ferent techniques.

I proposethat an expert probleni~solver~houlclbeableto employ multiple representationsfor the same problem.Within each representationradically different reasoningtechniquescan be used. By employing the differentrepresentations,the problem-solvercan solve problems ofvarying difficulty and, more importantly, use only thosereasoningtechniqueswhich areappropriateto the chilficufty ofthe problem. By definition, such a problem-solvermeets theabovecriterion.

in mechanics,a usefuldistinction can be madebetween qualitative and quantitative knowledge. Qualitativeknowledgerepresentsthe scenein termsof grossfeaturessuchthe generaltype of curveand the relative heights betweenpoints. Quantitativeknowledgerepresentsthe same scene intermsof mathematicalequationsdescribingthe shapesof thecurves and the numericaldistancebetweenpoints. A simplcqualitative rule usesthe relativeheights to determinewhe:heran object releasedat one point can reach the Other.Quantitalive reasoning,on theother hand,symbolicallymanipulatesthe mathematicalequationsand numericalquantitiesto obtain thesameresult.

A problem-solver which employs these tworepresentationsfor solving thesamr problem hasa number ofdistinct advantages:(I) It solves simpler problems with drastically simplertechniques.(2) Evenwhen qualitativeanalysisfails, it setstip specificplanswhich greatlysimplify the quantitativeanalysisof the problem(3) Qualitativeanalysiscan handle indeterminaciesin problemspecification.The first advantagehasalreadybeendiscussed.

Even if we were only interested in problemsrequiring equations,the qualitativeanalysisstill performsacrucial role in the problem-solving. By itself a mathematicalequationcontainsno useful information. To makeuse of anequationthe meaningsof each of the variablesnitist bespecified and the conditionsof his applicability mustbe known.Thereare manyequationsdescribingthe motion of movingobjectsandrelating thedimensionsof physicalobjects,but howdo we determinewhich of theseequationsare relevant to theproblem at hand? Although the qualitativeanalysisof theproblem may fail, requiring quantitative analysis. thequalitative analysis determinesthe kind of event happeningthus providing a concisestiggestionas to which equationsarerelevant.

The qualitativeanalysisalso providesan oveiallstructurefor the solution of the problem. A problem c~ininvolve a number of independentparts each requit lug

quantitativesolution. Considerthe problemof a block slidingovera hill first you must determinewhetherthe block canmake it to the lop of the bill, and then you must det~rmninewhetheror not the block flies off the hill becauseit is too steepon the other side. The qualitative analysisfirst presentstheproblem of reachingthe top for quantitativean.lysis, and ifthe top is reachableit presentsthe problemof whethertheobjectfalls off the hill on theotherside.

The qualitative argumentdoes not require acompletely describedscene. For example,qualitativeanalysistells you that a ball will roll down an inclined plane without“6: rt~ Kicer

needingto know the slopeof the incline or the radiusof theball. To determinethe velocity of the ball at the bottom of theIncline theseother quantities must be incorporatedintoequationswhich aresubsequentlysolved for the final velocity.

A difficulty introduced by multiple representationsiscommunication. One problem is that the way onerepresentationrefersto a particular entity is often radicallydifferent from how the other representationsrefer to this sameobject. Another problem is the format of the queries andrepliesbetweenrepresentations

Although the theory suggeststhat multiplerepresentationsareuseful, it doesnot provide any informationabout the detailsof the representationsor how to utilize tliemEn orderto explorethe theoryfurther an expertproblem-solverNEWTON has beenconstructed. It solves problemsin themechanicsmini-world of roller coasters’~(the kinematicsofobjectsmoving on surfaces). NEWTON is not intendedto hea generalmechanicsproblem-solver,but is only used todemonstratethe aboveprinciples. Within its mirii-woilct it catshandlea wide rangeofproblems. It usesqualitativeaugumentswhen possible,but will resort to equationsif necessaiy. Itrecognizesnonsensicalproblems. It doesnot becomeconfusedor ineffident when given irrelevant facts,objects or equations.

One extremelyImportant pieceof qualitativeknowledgeis theability to roughly predictwhat will happen ina given scene. For example,qualitativeknowledgeteil.s youthat an unsupportedobjectwill fall or that a rubber bali willbounce. We will call this envisioning. More formally.envisioningmeansgeneratinga progressionof snapshotswhichdescribewhat might or could happen. The most importantfeatureof a scenein the roller coasterworld is the position ofthemovingobject, andso the snapshotis bestdescribedby theposition of theobject. An entire event is describedby a lice,eachnodebeinga possibleposition for the objectandeacharcindicating an actionwhich movesthe object from one positionto another.

The quantitativeknowledge uses a FRAME-likeorganizationto packagetogether mathematicalequations.There are an extremely large number of different equationsthat could be applicableto any problem. Fortunately,sinceequationstend to come in groups,an individual decisionneednot be madeaboutthe relevanceof everysingleequalion. Therelationshipsbetweenthe anglesand sides of a sriingle loinsone suchgroup or FRAME. Another possible FRAME IS tit€kinematicequationswhich hold for uniformly acceleiatinbobjects. With this representationonly a small number ofdecisionsare required to determinewhich equationsaierelevant. It also providesa convenientway to providemeaningto the variables in the equations. Sincethe FRAME onlyapplies when certain conditions are met, the variables in theequationsof the FRAME can refer to these conditions toprovide a meaningto the variableswhich is generalamong allthe FRAMES.

In order to actually solvemechanicsproblems twoother distinctly different representationsbecomeimportant.One of the representationsoriginates from the qualitativeknowledgeand the other originatesfrom the quantitativeknowledge. The envisioning can often solve the problemdirectly, however, if it fails it must be able to articulate itsdifficulty for the FRAME-representations.This requutescarefullyanalyzingand transformingthetreesgeneratedby theenvisionmentutilizing separateanalysisand transformationrules. This analysisdetermineswhich top-level FRAISI F usrelevantto theproblem and which variablesneed to he solvedfor. Theresultof instantiatingthe FRAMEs is an informationstructurewhich relatesall the equationsand variablesrelevantto theoriginal problem. This dependencynetwork is thc’nanalyzed to identify possible paths to the solution. Symbolic

mathematicaltechniquesarethen applied to checkthe possiblepath. If the manipulation is unsuccessfulor intractablewiththesymbolic techniquesavailableother pathsaretried.

With theseadded refinementsthe organizationofNEWTON is as follows:

Envisioning representsthe original scenein termsof its grossfeatores detcimining what might possibly happen .andrecording this information in a tree. Its knowledge isrepresentedby rules which look at grossfeaturesand thenextendthe envision~ingtree. The remainingqualitativeknowledgerepresentsthe original problem in termsof the treegeneratedby the envisionment.Its knowledgeis representedastransformationand analysisrtiles whoseapplicationeventuallyresults in a plan to solve the problem. The quantitativeknowledgeis representedin FRAMEs, a small subsetof whichis instantiatedfor any particular problem. It olaintains aCONNIVER-like [McDermott & Sussman74) data-basetorepresentthe semanticsof variables. After instantiation theFRAMEs areexaminedfor possible pathsto the solution, andsymbolic techniquesare invoked to evaluatethese possiblepaths.

The only entities which can be referred to inmessagesbetweenrepresentationsarephysicalobjects, instancesof time, and variables describingthem. Every representationrefersto theseprimitive entities in thesameway. The problemof the format of the messageshasbeenresolvedby havingspecialptirposeexpertsbetweeneach two representationsthatever wish to communicate.This is not as unsatisfactoryas itmight seem sinceour theory imposesvery rigid constraintsonthe nature of tins wmmunicatton. A query is generatedonlyafter the problem is discoveredto be unsolvablein onerepresentation,and thus the query asks a very preciseand

ecu icted cluestion. The answersgenerated by queriesarelimited by the kindsof questionsasked.

- To get an overall idea of the differentrepresentationsconsiderthe following simple mechanicsproblem:

A small block slides from restalong the indicated frictuonlesssurface. Will the block reachthe point maiked X?

The following is a possibleprotocol of a humansolving the problem.

“The block will start to slide down the cutvedsurfac. without falling off or changingdirection. Afterreachingthe bottom it startsgoing up. It still will not fall off,but it may start sliding back, If the block ever reachesthestraight sectionIt still will not fall off there,but it may ueverseits direction. To determineexactlywhetherthe block reachesX

E~LVIS~OWING

ç~u(\IS[1\TIV E~

Q\F~NTIThT I ‘LIE

Cl S3

S2.

Cl-

Ca

Knriwlertgc’ Renr,—6: de Kleer300

CI

(2 g A1

)t 12

we must study the velocity of theblock as it moves along thesurface. The velocity at the bottom can be computed by usingconservation of energy:

Similarly, this velocity and conservation of energy can be usedto set up an equation which can be solved for the velocity (112)

at thestartof the straight section:

112 in V2~ 1/2 in v~2- nigh2

if’ thesolution for v2

is imaginary, then the straight segment isneverreached. At thestraight sectionwe can usekinematics tofind out whethertheblock everreachesX. Theaccelerationofthe block along the surface must be:

a gsin TThelength of the straight segmentis 1.. 1 cos T. so by the wellknown kinematic equation relating acceleration,distanceandvelocities:

03 v2

2- 2 L g tan T

Again if v3

is Imaginary, X is not reachable.”The first part of’ the protocol which involved

identifying a possible path to reach X we call envisioning.Envisioningdescribes(andso limits) theproblemspacelot mcclwhen we ignore the specific valuesof the variables. TheproblemhadvariablesA

1, A

2, T andL which wererequiredfor

the solution but the protocol (before the decisionto calculatevelocities) held true for a wide rangeof valuesfor thesevariables.The reasoningdependedon A

1> 0, A2 > 0. L > 0, 0 <

T < 90 andthe facts that all the curveswere concavelions theperspectiveof the object and that the curves weredifferentiable everywhere. All this information could beassumedfrom the diagram. Everythingthat was predictedbytheenvisioningwas achievablefor somevaluesof thevariablesand every possible assignmentof valuesto variables wasdescribed.

The envisionmentNEWTON generatesfor thisproblemis describedby the following figure. Theobject startsat corner CI, slidesthrough segmentSI, reachescOrner C2,slidesthroughsegmentS2, either slidesbackon segmentS~orreachescornerC3,andso forth. -

Many questionscan be answereddirectly from theenvisionment. For the aboveproblem the question “Wiil it

reachS2?” can be answeredwithout ftirthem reasoning.- Envisioningfails to answerthe questionwhen it predictsanumber of possibilities. When the block was sliding up thehill, it could not be determinedwhen or if it would start slidingback. ft is in identifying these multiple possibility points thatthe envisionersets up specific subproblenss. Even whenfurther reasoningIs required to resolvesuch a qualitativeambiguity,envisioning identifies thosepossibilities it mustdistinguish between, Although there was the problem ofdeterminingwhether the block would or would not slide backon the curve, the possibility of the block falltng off had beetseliminatedby envisioning. In summary,envisioninggives localand very specificproblemsfor further analysisand,on a globalscale, envisioning providesan organizationand plan to solvethe entire problem. The traceof the possibilities through timeprovidesthebasis for sucha plan. -

Quantitativeknowledge is used to disamhiguiatebetweenthe possibilities occuringat each fork. It is importantto note that FRAMEs are not procedures,but describedependenciesand assignmentsbetween variables. Thesedependenciesaresearchedfor solutionsto the goal variable. AFRAME is first examnined to determinewhetherit has thesolution for thegoal variable. If this seamcli is unsuccessfulallequationswhich referencethe goal variableareexamined,anda subgoal is generatedto find the remaining unknownvariablesin theseequations. All thecob FRAMEs referencedby the FRAME areeventually included in this search. Viewedfrom another perspective,the instantiationof a top-level

CL

FRAME resultsin a and-orgraph which must beanalyzedfoipossible solutions. Each equation which iefetencesthe goalvariable contributesto a disjunction since eachsuch equationcould possibiy yield iii ~olutio~. Unknown variables inequationsreferencing the goal variable contribute tocon junctions since every unknown in the equation inSist bedetermined to achieve the goal.

The MASS-MOVEMENT FRAME knows aboutmovements of objects on surfaces, but is not concerned aboutthe possibility that the objects may fall off the surfaces. Thefollowing is a descriptionof MASS-MOVEMENT and theother FRAMESit references. A FRAME usestwo kinds ofvariables: the names of the objects it is concerned about andmathematicalvariablesdescribing propertiesof theseobjectssuchas velocityor acceleration.

FRAME inaso--movernent (IF object sturface U t2VARIABLES:

(a ACCELERATION OF object,theta ANGLE OF surface)

IF surface IS flat THEN(USE right—trianqhe ON our face,USE kin ON object surface ti t2,a 6 oin(theta)),

USE energu ON objoct surface ti t2.

FRAME ener-qq OF object surface U 12VAR! ABLES;

(vi VELOCITY OF object AT TIME ti,vi : VELOCITY OF object AT TIME 12,h : HE I CHT OF surf ac:e)

vf2

— v12

2 G Ii,.

St

F~t~S-S.,

FAU.

Knowledge Repr, —6: ne Fleer301

FRAME right—triangle OF triangleVARIABLES:

Uibasehgpthetaltheta2

hyp sqrt(h2

sin ( the t allsin(theta2)

FRAMEkin OF object surface U t2VAR! ABLES:

(vf VELOCITY OF object AT TIME t2,vi : VELOCITY OF object AT TIME ti,d s DISTANCE OF surface,t t TIME BETWEENti AND t2,a ACCELERATIONOF object)

vf”’vi +at -

vf2 v12 ÷ 2 a d

d vi t + ~S a t2

.

NEWTON would solve this problem withoutFRAMES by comparingthe heights of the endpointsdirectly.In order to demonstratethe quantitative knowledge, thispossiblesolution (among many others)will be ignored. TheFRAMEs NEWTON uses are much more sophisticatedthanthese presentedhere; they are not shown since they involvemany featureswhich arenot relevantto this presentation. Inorderto get a betterunderstandingof how FRAMEs interact inthe problem-solvingprocesswe will artificially make Iii. A2. Tand L unknownandonly provide their valueswhen necessary

Envisioning determinesthat there is never apossibility that the objectwill fly off. Sincethe envisionnierittree for this problem hasa fork at S2 andS~t,the problem isdecomposedinto the two subproblemsof first disambigtiatingwhat happensat S2 andthen disambiguatingwhat happensatS3. This is the plan the qualitative knowledge developsforsolving the problem. In order to analyzewhat happenson S2,the velocity of theobject at the beginning of 52 must bedetermined.The velocityat the beginningof S2 is the sameasat the end of SI. so the first problem to be solved by thequantitativeknowledgeis to find the velocityat the endof SI.

To find the velocity at the end of SI MASS-MOVEMENTfiiOSt be invoked:

(MASS—MOVEMENT(81 SI TIMEI TIME2})Before MASS-MOVEMENTis invoked, vartaFtcsmust beassignedvaluesand meanings:(VELOCITY 81 TII1E1) -known(VELOCITY Bi TIF1E2I -desiredgoalWhen MASS-MOVEMENT is invoked IFRAME attemptstofind a valuefor this variable. Therearetwo placesin MASS-MOVEMENT where possibleassignmentsto VF take place.IFRAME discoversthat the assignmentin the conclitiotualcannotbe reachedsince SI is not fiat, The only alternativeistheENERGY FRAME. Using ENERGY is tinsuccessfulsinceHEIGHT is unknown, Every possibleattmpt ro achieve avalue for YE having now failed, the altcrnativeis to generardsubgoalsof discoveringthe variableswhich areblocking asolution to the desiredgoal variable. The path to \‘F isblocked by HEIGHT, but thereare i~oother accessiblereferencesto HEIGHT in ENERGY or MASS-MOVEMENTProblem-solvingnow haltsuntil HEIGHT is given,after whichIFRAME reexaminesENERGY and returnsa value for yE.This value is rememberedand thesegmentS2 is examinedinmuchthe sameway using the YF on Si as VI on S2:

(MASS-MOVEMENT (81 92 TIME2 TIMC3))Note that ENERGY returns an “impossible” result if theequationsresult in an imaginarysolution, thus indicating thattheobjectcannottraverseS2. -

On S~IFRAME has two possiblepathsto a solution

If ENERGY is tried it fails becauseHEIGI-IT is unknown.SinceS3 is flat KIN can be tried for a solution. For KIN iosucceedeither D or T must be known. Again everypath to VFIs blocked. Finding a value for either HEIGHT. D or 1’ wouldbe sufficient to solvefor VF. Thereareno other referencestoT in the FRAMEs so T cannot be achieved in MASS-MOVEMENT. The two variablesHEIGHT and D can hefound by theFRAME RTRI. RTRI is then invoked onsegmentS3. Thereis not enoughinformation to solve for thecevariables. The valuesfor TI (angleof surface)and L (baselength of surface)in the instantiationof RTRI on S~aie givenand IFRAME proceeds. Now RTRI returns with values forboth D and HEIGHT. IFRAME hasa choice betweenreexaminingK IN or ENERGY to solve the problem.Dependingon whether this results in an “impossible” solutionor a particularvalue for YE, the questionof whetherC’! usreachablehasbeetsanswered.

In order to demonstratethe importance ofenvisioning and to illustrate the natureof the omnsuiiicatioisbetweenrepresentations,a simplified version of envisioningand its Interaction with quantit?.tiveknowledge will hediscussedin the next section. For a more completediscrissionof this topic and the otherrepresentationsthe readeris referredto ~deK leer 75].

The most basic and primitive knowledgeaboutphysics is envisioning. For NEWTON, envisioning meansgeneratinga progressionof scenesencodedin a symbolicdescription which describewhat could happen. Both inNEWTQN andl in people,the envisionmentof the event is thefirst step in the understandingof theproblem.

In fact, envisioning is necessaryto understandtheeventat all. To understandthat a pencil might roll off a tablerequiresthe ability to envision that event without actuallyseeingit happen. Envisioning is pre-physicsknowledgeand itspresenceis independentof the goal of solving mechanicsproblems. Envisioning is, however, the fundamentaltool forunderstandingmechanics.

The envisioner requiresthat the original path bedescribedin termsof segmentsfor which the slopeandconcavitydon’t changesign, and achievesthe envisionmentbyapplying general rules which describeobject motions onsegmentswhosemonicavity arid slopedon’t chnr~y,esign. Ac aconsequencepoints at which the slopeor concavity arezero ordiscontinuousarc’ identified. In NEWTONenvisionmentstaitswolt the initial situation, identifies what is happeningin thatsituation,generateschangedsituations,andrecursivelyanalyzesthesenew situationsuntil all possibilitiesareexhausted. It mayseemthat it would be more efficient to only explore thesItuations that lie on some path to thegoal [Rundy ‘i’ll.Unfortunately, this goal directed reasoningrequiressomemeasureof’ how close it is to the goal, and this requiresaseparateanalysisof the scene. An extra pre~analysisof thesceneto identify pathsto the goal (nearnessto goal) roustinvolve as much work as theenvisioning itself, Simplestrategiessuch as ‘when the goal is left, move left” areinsufficient when thepath turnsaround.

The envisionmentresults in a tree,every foi k ofwhich indicatesa qualitativeambiguity which needsto beresolved. ‘rite forks for the roller coasterdomain can heclassifiedInto theactionsthatoccur there:

HEIGHT OF triangle,BASE OF triangle,DISTANCE OF triangle,ANGLE1 OF triangle,ANGLE2 OF trian~le}+ base

2)

h / hypbase / hyp~

OR

SLiDE-SLIDE FLY-SLIDE

Knov,ledge Repr~—6: etc Fleer302

~G

FLY-SLIDE-SLIDE TRANSFORM-TRANSFORM

For eachkind of qualitativeambiguity thereexistsatop-level of FRAME which can be used to resolvethatambiguity. Many of the FRAMES for different ambiguitiesreference each other. Wehave already seen one such collectionof FRAMEsfor SLIDE-SLIDE: MASS-MOVEMENT.

After the envisionment is completed, NEWTONaccumulatesa list of ambiguitieswhich must he resolved toreach the desired goal. This primitive plan is then analyzed tooptimize the disambiguationsthat must be done. The mainoptImIzation Is the deletion of plan steps. Plan steps can hedeletedif succeedingplan steps implicitly repeatit. The planfor theexamplesliding block problemIs:

SolveSLIDE-SLIDE on S2 and fail if C3 is not reached.SolveSLIDE-SLIDE on S~tandfail if X is not reached.

The secondplan step completely determinesthe relevanceofthe first step to the goal. Hence it can be eliminated. Someothereliminationsare:

SLIDE-SLIDE, FLY-SLIDE-SLiDE, FLY-SLIDE -> FLY-SLIDE-SLIDF

SLIDE-SLIDE, FALL-SLIDE, FALL-SLID E(symmernica I) ->

FALL-SLIDEEnvisioning looks at only one point or segment at .a

time and the possibleactions it determinespertain only to thatparticular point or segment. Sincethe envisioner titus localinformation to determinelocal actions, the primitive plait it

generatesis restricted to employing local quantitative techniquesWith the help of these transformation rules which eliminateunimportant local plan steps, the important ‘global structureofthe problem can be identified. The resulting plant can takemaximal advantageof the global quantitativetechnique ofconservation of energy.

Problem Formulation

NEWTON contains quantitative techniques tohandle every possible qualitative ambiguity which can hegenerated by the envisioner. Given the absolutenumericalposition of every point and the shapeof every segment,NEWTON can alwaysdeterminewhat will happenif it less thc’FRAM Es to deal with the segmenttypes. There are nianyother qualitative questionswhich can be asked besides“What

happens?”The qualitativeformulation of a problem leaves outmany details in both the description of the scene and thequeiyabout that scene. NEWTON attempts to reducequestionstothe simple “What happens?” type as much as possible.NEWTON also has some techniques to deal with otherquestion types,but theseareseverely limited and it is difficultto characterizewhatclassof problemsthey solve.

The question can involve a quantitative requestabouta qualitative predicate: “What is the velocity of theobjectwhen it falls off?” NEWTON firtt tries to satisfy thequalitativepredicate,andif that predicateis satisfiedattemptsto determinethe velocity when it falls off.

If the original problemcontainssymbolic patameters,and all the qualitativedisambiguationscan be made,it returnsthe final result in termsof theseparameters.NEWTON failswhen an unknown parameter makes a qualitativedisambiguationimpossible. (A smartersystemmight give adisjunctiveanswer.) If NEVu1TON is provided with an explicitnumericalrangefor a particularparameter,it will attemptto dothe disambiguation with this limited information.

A problem can specifya resultanteffect and askwhat Initial conditions lead to that effect. NEWTONsolvesthis type of’ problem by hypothesizing variables for all therelevantinitial conditionsand the comparisonswhLh otherwisewould be used in disambiguation are used to accumulateinequality constraints on theseinitial conditions. ‘Ihe loop-the-loop problem is anexampleof this type:

A small block starts at rest and slidesalong a fricticinless loop-the-loop as shown In the figure. What should the minimalinitial height be so that the block successfully completestheloop-the-loop?

The problemcan implicitly refer to pointswhich arenot present in the original figure. NEWTON can introctricepoints in a figure which arenot zeroesor sitagulanities.

A small block slidesfrom rest from the top of a frictionlesssphere. How far below thetop doesit lose contact with thesphere?

There are many questionsabout the roller coasterworld that NEWTON cannotsolve:“What is the period of an objecton a cycloid?”“Would the object still make it if the height of the hill wasincreased?”“What is the curveof shortesttime betweentwo points?“If the fall-off’ velocity is determinedwhat must the initialheight be?”

Concluding_Remarics

We have seen how NEW’ION can solvea widerange of problems in its mini-woild. The power of thetechniques it rises are appropriate to the difficulty of theproblem it is asked to solve. When qualitative arguments willwork it rices them, and otherwise nesotts to mathematicalequations. The organization of multiple representations allowsit to limit the calculations as the problem-solving progresses.

Envi skin in g is an extremely powerful toot forproblem-solvingand deservesfurther attention. In order to doany probh-m-sotvingabout moving physical objects some kindof’ envisioning is required. Cleatly, an envisioner is needed its

FALL-SLIDEF\Li.

lKnowl edge Repr,—6: 1e Fleer303

order to build a general mechanicsproblem-solver.Envisioning would be useful in other donsains such aselectronicswhich dependon mechanicalintuitions. In general.envisioning might be useful in every problem-solvingdomainfor which models exist. An example of a domain wheteenvisioning plays no rote is pure mathematicsor the solving ofequationsindependentof somedomain.

NEWTONhas three weaknesses which should hethe subject of future researchin this area: a lack of a themyof’ question types, an insufficiently powerful envisionet, and aninappropriatetheoryof mathematicalexpertise.

The current envisioner is relatively simple since itworks with a single point object in only one dimension. Moregeneral envisioning requires the manipulation of’ more thanone object at a time and an analysis in more thiu,n onedimension. Font [76J describes a program WHISPER whichcan do certain kinds of’ two dimensional envisioning. He ismore interestedin exploring theuse of analoguesthanproblem-solvingin general. A slightly extendedworld forNEWTON would bea world with two moving objects. Thedifficulty here is that theenvisionenneedsa notion of time. Anaive solution would be to envision the movementof eachobject independently, and then assume a possible collision ateach intersecting point. Each such point again ~ieldstwo newtrees. Two dimensional envisioning is far more complex sinceit introduces a large variety of new objects and possibleinteractions between them. Consider the problemof computingtheperiod of this pendulum:

Two dimensional envisioning is needed to see that the nailshortensthe length of the pendulumstring thus shortening thethe period.

NEWTON has no good theoryof question typesNEWTON can solveall “What happens?”questionswhich sictotally described,and fortuitously slight modifications of thesetechniquescan solvemany other problemtypes. Unfot tun.ately,these other problem types cannot be characterized very well.There is a need for a more powerful theory of question types.and techniquesto dealwith these.

One of the unexpecteddifficulties eneoiuriteresi iii

implementing NEWTON was the interaction betweenquantitative knowledge and mathematical expertise.NEWTON’s mathematical expertise is puovided by routinesculled from MACSYMA. The problem in using these routinesis that they are just black boxes, and the quantitativeknowledge needs other kinds of interactions than thosenormally provided by theseroutines. A simple example is theoccurrenceof multiple roots. In general MACS\’MA doesnotgenerate appropriate explanations for why it fails to achievesome particular manipulation. NEWTON should have hadanother representationbetween the quantitative and themathematical which knew about mathematiss and aboutMACSYMA. -

Thesedifficulties originate from a far naore seriousshortcoming of NEWTON: it treats equations purely asconstraint expressions. It Is solving a physical problem forwhich an inherent duality exists betweenthesymbolic structureof the expressions and the actual physical situations. Eachmathem.,tucal ituanipulation of the equationsreflects sonicfeatureof the physical situation. The manipulation should beunderconstantcontrol and observationsuch that any unusualfeatures or difficulties should be immediately reported to therest of the problem-solverwhich then decides how to proceed.This requires that theproblem-solverhave much more contiotover the symbolic manipulationroutinesand constantlynaonitortransformationson the expressionsto what import they have

on the physicalsituation. Onepossibility for the nianiptilationroutinesare thosesuggestedby Bundy EBtindy 75).

Acknow1e4g~ments

This researchwas conductedat the Artificial IntelligenceLaboratory of the MassachusettsInstitute of Technology.Support for the Laboratory’s artificial intelligence researchisprovided in pant by theAdvancedResearchProjectsAgencyofthe Departmentof Defenseunder Office of Naval ResraichcontractnumberN000I4-75 C-0643.

References

[Bobrow, 68)Bobrow, CD., “Natural Language Input for a ComputerProblem SolvingSystem”, in Minsky (ed.), SemanticInj’ors’ielionProcessing,Cambridge: M.I.T. Press,1968.

[Bundy 75]Bundy, Alan, ‘Analyzing MathematicalProofs (or zeachingbetween the lines)”, Department of Artificial Intelligence,ResearchReport~2,Universityof Edinburgh.1975,

tEondy 77)Bundy, Alan, “Will it Reach the Top? Prediction in theMecnanics World”, Department of Artificial Intelligence.ResearchReport *31, Universityof Edinburgh,1977.

[Charniak68]Charnlak, E., CARPS, “A Programwhich solves CalculusWord Pro’hlems”, Project MAC TR-51, Cambridge: M.I.-~.,1968,

[de Kleer75)de RIcer, ,Johan,“Qytalitative and Ojiantitative Knowledge iii

ClassicalMechanics”,Artificial IntelligenceLaboratoryTR-352,Cambridge: M.I.T. 1975.

tKleppner& Kolenkow 73)Kleppner,Daniel, andRobert j. Kolenkow, An Introduction toMechanics, New York: McGraw-Hill, 1973.

[Mathlab 74)Mathlab Group, “MACSYMA ReferenceMatinal”, CaniU~ucige.M.I.T., 1974.

(McDermott& Sussman‘/4)McDermott, Drew, and Gerald Jay Sussman,“The ConniverReference Manual”, Artificial Intelligence Laboratory. AIM-259a,Cambridge: M.I.T., 1974.

[Minsky 73)Minsky, Marvin, “Is Framework for RepresentationofKnowledge”, Artificial Intelligence Laboratory, AIM-SOS,Cambridge: M.I.T., 1973.

tNovak ‘76)Novak, Gordon Shaw, “Computer Understandingof PhysicsProblemsStated in Natural Language”,The University ofTexasat Austin, Ph.D.Thesis,1976.

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