Caricatures: Generating Models of Dominant Behavior

Olivier Raiman,Brian C. Williams*Xerox Palo Alto ResearchCenter

3333 CoyoteHill Road, Palo Alto, CA 94304 USA

Abstract

When analyzing most physical phenomenathecomplexityof the correspondingequationsrapidlybecomessuch that there are no generalmethodsto derive exact solutions. A solution prevalentinacid-basechemistryis to constructasimplified so-lution that preservesonly the dominant behav-iors. Since what dominatesvaries, for example,dependingon the strength of the acid or its ini-tial concentration, the chemist divides behaviorinto a patchwork of simpler subregimesthat re-flect thesevariations. The successof suchan ap-proachhingesupon thecareful identificationof thesimplifying assumptions(e.g., the acid is strong)which induce thepartition. What is moststrikingis that theseassumptionsappearto ariseprior tothoughtsabout how the modelsareto be usedmodelling is an emergentprocess.

To identify dominant regimes we exploit themetaphorof a caricature — an exagerationofan equation’s prominent features— to generatethe requisitesimplifying assumptions.Generatingthese assumptions,the boundaries of the parti-tion, and thesimplified equationsfor eachregime,draws heavily upon our earlier work on qualita-tive algebraicand order of magnitudereasoning.The resulting process,called caricatural modeling,is sufficient to replicatea broad set of examplesfrom acid-basechemistry.

IntroductionAlong with otherswe haveargued [0, 0, 0, 0] that themodel generation/selectiontask is ultimatelydriven bythe phenomenaof interest,asdictated by the problembeingsolved. Thus, we found it striking, when exam-ining analytical chemistry texts[0], that much time isdevotedto teachingmodelingskills at thestart — withno mention of how thesemodels are to be used. Webelievethis is aninstanceof anovelandpervasivefacetof modeling that requiresexplanation.

~The orderingof authorsis incidental.

We claim that modeling is often an emergentphe-nomena— thereexists adistinct notion of interestingphenomena,which is not contingenton the task beingperformed. The argumentgoesas follows: Peopleareextremelyinventive; theyarequite good at makinguseout of just about anything they understand,whetherit be the designof a mechanicaldevice or a chemicalsynthesis.’ The difficult issuethen is to comeup withmodels that, while accurate,are sufficiently simple tobe intuitively grasped. Thus, at least for invention,the problem of constructing simple, but accurate mod-els may precedeuse.

We observethat emergentmodels achievesimplic-ity, by highlighting dominant behaviors, and carvinga system into a patchworkof regimeswhere differentbehaviorsdominate. Finally, we claim this patchworkemergesascaricaturesof the system,by reinforcing itsprominentfeatures.We presenta domainindependentapproachto generatingemergentmodels, called can-catural modeling, anddemonstrateit in the context ofacid-basechemistry.

An examplefrom chemistryConsiderequilibrium behaviorof asimple reaction —

the dilution of acid molecules,All, into water. Thedilution is characterizedby the reactions1120 — H~+011 and All H~+ A, andits equilibrium stateisgovernedby:

Ca

K,, ah

where ah, a, h+, h,o, and oh denote theconcentration2at equilibrium of the speciesAll, A,11±, 1120,and011—, respectively,K~and K,, areequi-librium constantsfor the water and acid ionization,

‘The approachof focussingforemoston making inter-actionstractableduringinventionwe call interaction-baseddesign.[0]

‘Although the concentrationof speciesS is tradition-ally denoted [5], this conflicts with the use of [1’ withinqualitative reasoning, to denotea quantity’s sign.

(Ii) Chargebalance:(12) Massbalance:(13) Water equilibrium:

(14) Acid equilibrium:

= oh+a,= ah±a,= h+oh,

= h+a_,

1

and Ca denotesthe initial concentrationof All. Com-ing up with these equations is straightforward; themost reasoningintensivestep is to solve for the equi-librium concentrations.

Bypassingbrute force through apatchworkof dominant behaviors

Toderive, for example,theconcentrationof 11+ ions wecould usea bruteforce approachto solving the systemof nonlinearequations. Eliminating ah, a, and ohin the four equationsyields the following equilibriumconcentrationequation:

h~3+ K,,h~ (KaCa + K~)h~= KaKw.

The derivation of this equation is used in chemistrytexts to make clear the need to avoid brute force.Even for this simple case, the concentrationequationis a third degree polynomial in h~,making it diffi-cult to solve(solving this equationinvolves computingthe real roots of the left hand side). For more com-plicated cases,such as polyprotic acids — acids withmore than one replaceablehydrogenion 11+, suchas113P04— the degreeof such equationsincreaseswiththe number of replaceableions. For example,H3P04has threereplaceableions and results in a concentra-tion equationof degreefive. Of course, there are nogeneralsolutions to algebraicequationsof degreefiveor higher (by Galois). Likewise, derivingeachconcen-tration equationinvolvessolving asystemof non-linearequations,which can be quite computation intensive.Thus, applying brute force reachesa deadend for allbut the simplest cases.

Instead, a chemist is taught to proceed as follows(takenfrom [0], chapter5). First, having introducedequations11—14, governingthe reaction’s equilibrium,the chemistguessesseveral interestingsimplifying as-sumptionsabout what the dominant speciesmay be:

The acid is weak (a <<C,,).The acid is strong (ah << C,,).The soln. is essentiallyneutral (a <<hf).The soln. is stronglyacidic (oh <<hf).

Combining, for example, assumptionsA2 and A4and applying them to the chargeand mass balanceequations(11,12) producesh~ a (Ii’) and C,, ~u(12’). Solving for h~results in h~~uC,,, afar simplerresult than producedthrough brute force.

Applying other combinationsof assumptionsin asimilar mannerproduces:

Al:A2:A3:A4:

Assump T{j~ Simplified ConcentrationEqn~A2,A4 J~F h~ u C,, (El)A3 R2 h~’ K,,~.A2 R3 h~’—C,,h~ K,~

A4 R4 h~’+K,,h~ C,,K,,Al,A4 R5 h~’ C,,K,,Al R6 h~’ r~ C,,K,,+K,,,

none R7 h+’+K,,h+’—(K,,C,, ±K,~)h~

= K,,K,~

The remaining step is to determinethe domain ofvalidity for eachset of assumptions— the constraintsthat the assumptionsimpose on the givens, K,,, K,,,and C,,. Returning to the pair of assumptionsA2 andA4, fromA2 (ah <<C,,) we deriveC~/K,,<<C,, by sub-stituting for ah and a using the simplified acid equi-librium (14), acidconcentration(El) andmassbalance(12’) equations. And from A4 (oh << h~)we deriveKW/C,, << C,, by substituting for oh and h~usingthe simplified acid concentration(El) andwater equi-librium (13) equations.Thesetwo constraintsdefine aregion, Ri, whosefringe correspondsto the two boldlines in the upper right corner of region diagram be-low (taken from [0], p. 75). The domainsof validityR2—R7for theremainingsetsof assumptionspartitionsthe reaction’s behaviorinto simpler regimesaccordingto the valuesof Ca and K,,. Given theseresults, theproblemof identifying asolution’sacidity for givenval-uesof C,, and K,, involves identifying the appropriateregionand applying the correspondingsimplified con-centrationequation. This paperdemonstrateshow toautomatethis style of reasoning.

C.)

0

-14 a<bwhen

For Kw = 10 a/b<0.1Elementsof the chemist’sexpertise

What arethe essentialcharacteristicsof the aboveex-ample? The intractability of solving the initial equa-tions is avoidedfrom the start by replacing them withsetsof equationswhich are simpler to solve, and arestill relatively accuraterenditions of the initial equa-tions. An accurate,yet simpler rendition is achievedthrough a pervasivestyle of reasoning: first the com-plex behavior is partitionedinto asetof regimeswheresubsetsof the behavior dominate; then the equationsareapproximatedby eliminating all but thedominant

-14 -8 -2log(Ka)

-2

-8

2

behaviorsin that regime. The resultingequationsareeasilysolved,circumventingthe needfor sophisticatedmathematics.

The essentialskill — which is deeply rooted in thechemist’sknow-how but poorly systematized— is theability to identify thedominant regimesandtheir cor-respondingsimplifying assumptions.We offer here anapproachfor identifying such regimes,which is suffi-cient to replicate a broad set of acid-basechemistryexamples(taken from [0]), and draws heavily on ourearlier work on qualitative algebraicreasoninginvolv-ing hybrid qualitative/quantitative [0] and order ofmagnitudealgebras [0]. This approachis basedonthe metaphorof a caricature, which providesclues towhat are the interestingsimplifying assumptionsandthe correspondingregimes. Its instantiationis a pro-cesswe call canicatunalmodeling.

What is a caricature?From acommonsensestandpointacaricatureof an ob-ject is a description which exageratesprominent fea-tures, andeliminatesinsignificant features.For exam-plecaricaturesof RichardNixon reducehis faceto littlemore than a nosewith an exaggeratedslope. Apply-ing this concept to modeling, givena systemof initialequationswe construct a caricature of this systembyexageratingone or more of the equation’sprominentfeatures. In this paper, we take “prominent” to meanthat one term a of an equation E dominatesanotherterm b: at > bi; that is, a is further from zero than b.We exageratethis featureby making a muchgreaterthan b, thus making a dominant and 5 insignificant:a > b -~-~ a >> b. We call this relation a caricatu-ral assumption. By using this assumptionto simplifyE we producea cancatural equation,which eliminatesthe insignificant features.

For example,given that all concentrationsare pos-itive, two prominent features of equation 12 (Ca =ah+a) are C,, > ah and C,, > a (note that allconcentrationsare positive). Exagerating C,, > ahintroducesthe caricaturalassumptionC,, >> ah, andallows 12 to be replaced by the caricatural equationC,, a. This correspondsto the chemist’s notionof a strong acid (i.e., essentiallyall All dissociates).Conversely, exagerating C,, > a introduces theassumptionC,, >> a, and produces the caricatureC,, ah, the chemist’s notion of a weak acid (i.e.,a negligible fraction of the acid All dissociates).

Of course an alternative approach might take aquantity or subtermfrom any two equationsand pre-sumeone dominatesanother. However, the numberof potential assumptionswould be prohibitively large.Insteadthe conceptof caricatureallows us to use ex-isting featuresof the initial equationsascluesto whatrelations are worth exaggerating. What is striking isthat the restricted set generatedthrough caricaturesmatchesthe simplifying assumptionsintroduced in avariety of acid-basechemistry examples.

There are two additional issues. First, more thanonefeaturemay be exagerated.For example,portraitsof Nixon often exagerateboth his noseandjowls. Like-wisewe might exagerateC,, relative to both ahanda.Second,exageratinga set of features may not alwaysbe consistent. For example,combiningboth caricatu-ral assumptionswith equation12 allows us to concludethat C,, >> C,,, which is inconsistentfor positive C,,.

Following the example,after identifying the caricat-ural assumptionsandgeneratingcaricaturesof the ini-tial equations,what remainsis to solvefor thesimpli-fied concentrationequationsand the domain of valid-ity for the caricaturalassumptions. As we elaboratein the next few sections,caricaturalmodeling involves1) generatingsetsof caricaturalassumptions,2) deriv-ing the caricaturesof the initial equations,3) solvingfor the simplified concentrationequations,4) derivingthe domainsof validity, and 5) recognizing inconsis-tent setsof assumptions. Spaceprecludesa detailedpresentationof the algebraicmanipulations,describedelsewherein [0, 0]. Rather,our goal is to demonstratehow, by exploiting thesealgebraic manipulationtech-niques, caricatural modeling is able to replicate thechemist’stacit skills.

Generatingcaricaturalassumptions

We begin by extracting the prominent features fromeach initial equation. We expressboth features andequationsusing the hybrid qualitative/quantitativeal-gebra SRi (an algebra combining signs and reals).Briefly, the domain of SRi extends the reals to in-clude signs (i.e., 4- (0, inf), (— inf, 0) and

(— inf, inf)). The operators of SRi extend thestandardoperatorsof the reals (+,—,x and /) to thislargerdomain, resulting, for example,in the combina-tion of areal andsign algebra. As usual[~-]mapsareal

to its sign. In SRi an inequality, suchas C,, > ah isexpressedby the hybrid equation[C,, — ah] = 4-.

Recall that a prominent featureof an equationis apartialorderbetweenabsolutevaluesof anytwo terms.Extraction is performedusing Minima’s algebraicsim-plification proceduresfor SRi, substitution of equalsor supersets,and for more complicatedexamplestheSRi hybrid resolution rule, asdescribedin [0]. For ex-ample, from the (quantitative) massbalanceequationC,, = ah + a (12) and a being positive (the qual-itative equation [a] = 4- (P1)), Minima derivesthehybrid equation[C,, — ah] = 4-, equivalentlyC,, > ah:

Si) C,, — ah = a Cancellationon 12.S2) a c [a] Definition of [].

S3) C,, — ah C [a4 Subst a in Si using S2.S4) C,, — ah C 4- Subst [a] in S3 using P1.S5) [C,, — ah] = 4- SRi simplification of S4.

It follows then from S5 and [ah] = 4- that IC,, >aht. Exageratingthis featureaccordingto

a > b a >> S

3

InitialEqns

h~= oh +a (Ii)h~=oh+a (Ii)

C,, =ah+a (12)

SignEqns

[a] = 4-,[oh] = 4-[oh]=4-,[a]=

[ah] = 4-,[a] = -4

prominent—__featureh~> oh

h~>IaC,,> Ia-I

caricaturalassumptions

h~>>oh (A4)

h~>>a (A3)C,, >>a (Al)

C,, = ah+ a (12)K,,, = h+oh_ (13)

K,,ah = h+a_ (14)

[a] = -4, [ah] = -4 C,,> ahlnone

none

C,, >> ah (A2)none

none

Deriving the caricatureof a regime

Given a set of caricatural assumptionsdefining asubregime, the order of magnitude algebra systemEstimates[0]is used to derive the caricatureof the ini-tial equations,the simplified concentrationequationsand the domain of validity. First, the assumptionsare used to simplify the initial equations,resulting ina set of caricatural equations. Caricatural assump-tions and equationsareexpressedin Estimate’sorderof magnitudealgebra. The types of dominancerela-tions usedearlier, a << b and a b, are capturedasalgebraicequationsin Estimates:a >> b S C ca, anda S a C (1 +e)b, wheree denotesaset of (positiveandnegative) infinitesimal values.3

Given a set of caricaturalassumptions,Estimatesproduces a set of caricatural equations, by applyingthe assumptionsto each initial equationusing orderof magnitudesimplification, substitution of supersetand qualitative resolution[0]. For example, considerthe pair of caricatural assumptions: C,, >> ah (A2)and h~ >> oh(A4), which correspondto the exam-ple at the beginning of the paper. Applying A2 to themassbalanceequationC,, = ah + a (12), Estimatesderivesthe caricaturalequationC,, a (12’) throughthe following sequence:Ti) ah C cC,, Estimatesequationfor A2.T2) a C —cC,, + C,, Subst ah in 12 with Ti.T3) a C (1 + e)C,, Simplification of T2.T4) a ~vC,, Relationequivalentto T3.

Likewise,applying A4 to chargebalanceh~= a +oh(Ii) results in h~iv a (Ii’). Applying theseassump-tions to 13 and14 providesno simplification.

Next theconcentrationequationsare derived. Givenan equilibrium concentration,such ash+, andthecar-icatural equationsjust derived, Estimates is used tosolve for h+ in terms of the givens K,,,, K,, and C,,,producing h~ C,, (El):

‘Intuitively eadenotesthesetof all valuesmuchsmallerthan a, and(1 + €)a denotesall valuesclose to a.

Equilbrium concentrationsfor ah,a andoh arede-rived analogously.

Finally, eachbound of thedomain of validity corre-spondsto one of the caricaturalassumptions,and isderived using Estimatesthrough a processsimilar tothe above. A boundary is derived from an assump-tion using the caricaturalequations(throughqualita-tive resolution)to eliminatetheequilibrium concentra-tions, resulting in aconstraintbetweengivens(K,,,KWandC,,).

For example,from Ca >> ah (A2) Estimatesderives

K,, >> Ca, using 14 (K,,ah = h~a), 12’ (h~ C,,),and El (C,, iva):Si) C,, >> h~a/K,, Resolveah in A2,14.S2) C,, >> h~C,,/K,, Resolvea in Sl,12’.S3) C,, >> C~/K,, Resolveh~in S2,Ei.S4) K,, >> C,, Simplify S3.

The bounds and concentration equations derivedthrough these processescorrespondexactly to thosein the exampleof section

As afinal note, in somecasesasetof caricaturalas-sumptions will be mutually inconsistent,for example,as we pointed out earlier for {Ai, A2}. This is recog-nized when oneof thecaricaturalequationsderivedbyEstimatesis not self consistent— for example,from{Ai, A2} EstimatesderivesC,, >> C,, — or is inconsis-tent with the inequalitiesderivedby Minima.

Creatingthe patchwork

The relation betweencaricaturesof different regimeshas a variety of interesting properties, somehavingimportant computationalconsequencesfor caricaturalmodeling. Interrelationshipsbetweensetsof assump-tions can be visualizedusingasubset/supersetlattice:

producesC,, >> ah, which is equivalentto assumption Ui) (1 + e)h+ — a D 0 Estimateseqnfor Ii’.A2 of section . The derivation of each featureand its U2) (1 + c)C,, — a C 0 Estimateseqnfor 12’.correspondingcaricature is summarizedin the above U3) (1 + e)h~— (1 + e) C 0 Resolvea in Ui,U2.table. U4) h~iv C,, Reln equiv to U3.

4

..-. —

{Al. X)~A3.A4 I

Al. .~. A3 } {Ai. .~tA4 (Al ,~, A4 } A2. .~. A4 I

~~

IAI} {A2} (A3} (Al

{}First, note that at the bottom the lattice is rooted

in theoriginal model — sincethereareno assumptionsno exaggerationhasbeenperformed. And as we moveupwardsthrough the lattice the models becomesim-pler, since eachassumptionmakesan additional terminsignificant, which then dropsout of the equations.

Second, although models higher in the lattice aresimpler, their domain of validity is more restrictive.Sinceeach caricatural assumptionintroducesa sub-regime boundary, the region correspondingto the do-main of validity of onecaricatureis a subsetof thosefor any caricatureappearingbelow it in the lattice.

Third, when moving up the lattice additional as-sumptions do not alwaysresult in simplification. Forexample,{A2, A3} producesthesameequationfor h+as does {A3}.

4 This explains why Schaum’soutlineincludes a region, R3, for {A2} but no region for{A2, A3} (see the region diagram of section ). Thesame argumentapplies to the absenceof {Ai, A3}.These eliminatedsets are depicted by squaresin thelattice. Likewise,additionalassumptionsdo not alwaysrestrict the domain of validity, in particular when theboundarythey introduceis outside theexisting region.

Finally, while all caricaturescould be generatedbysimply repeatingthe approachof the previoussectionon all combinationsof caricaturalassumptions,the dif-ferent combinationssharetwo propertiesthat can beexploited to make this processmore efficient. First,by monotonicityeachsupersetof an inconsistentsetofassumptionsis alsoinconsistent. Thusto avoidexplor-ing potentially largesectionsof the lattice, we createcaricaturesstarting at the bottom of the lattice andmovemonotonicallyupwards,ignoringanythingabovean inconsistentset. In our example,of 16 potentialsetsof assumptions,9 prove consistent,2 are explic-itly demonstratedinconsistent,and5 are supersetsofthese, and thus neednot be explored. The 7 incon-sistent setsare markedby X’s in the lattice. Finally,caricaturescan also be generatedincrementally by ex-ploiting monotonicity. Given thecaricatureC for asetof assumptionsS (in particular C containsthecaricat-ural assumptionsandcaricaturesof initial equations),the caricatureof its immediatesupersets,S U {A} arecomputedby further exageratingC using assumption

4But this dependson how many of the equilibrium con-

centrationswe are interested in. {A2,A3} may allow addi-tional simplification over {A3} for other species.

RelatedWorkAn important distinguishing featureof our work, hereandin [0] is theemphasison modelgeneration. In con-strastto the perspectivehereon modeling asan emer-gentprocessandthe focus on quantitativedescriptionsof dominant behaviors,critical abstraction [0] focuseson extracting qualitative featuresof modelssufficientto understandabehaviorof interest. Our ultimate goalis a generativemodelling approachthat bridges theseextremes.

Thereis a largebody of complementarywork on theproblem of selecting betweenexisting models, whichcan exploit the models that generativemodelling cre-ates. [0, 0] use a graph of model relationshipsandsensitivities to help select amongmodels. [0, 0, 0, 0]selectappropriatemodels for the constituentsof a de-vice, exploiting informationsuppliedabout simplifyingassumptions,thequantities beingobserved,and in thelast two casesthe desiredaccuracyof the approxima-tions.

Additionally, Caricatural modeling demonstratesthe power of qualitativealgebraicskills, in particularthe useof Hybrid qualitative/quantitativealgebra toreasonabout critical features,andorderof magnitudealgebra to reasoningabout dominance. Severalothertechniquesmay be applicable to these two subtasks[0, 0] and [0, 0].

Third, [0] explores the idea of partitioning statespaceand approximating behavior through a set ofpiecewise linear approximations. A concern here isthat linearizationthrows awaysomefeaturesof behav-ior that are particular important, such as the geomet-ric coupling that results from the product of two vari-ables. By concentratinginstead on dominant behav-iors, caricaturalmodeling avoidsthis limitation whilestill achievingsimplicity. The ideaof exagerationhasalso beenapplied to simplifying the DQ analysisprob-lem, as exploredby [0, 0].

Finally, two piecesof researchon acid-basechemistryarerelevant: [0] usesEstimate’sorderof magnitudeal-gebrato simplify equilibrium equations,but doesnotcapture what we find most interesting, the generationof the caricaturalassumptionsand the patchwork ofregimes. In contrast to our focus on equilbrium be-havior, the kineticist’s workbench[0] extractsfeaturesof the dynamicsof chemicalreactions.

AcknowledgmentsWe would like to thank Johan de Kleer for feedbackduring our hour of need.

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