Qualitative superposition

Artlficml Intelligence 56 (1992) 171-196 Elsevier

171

Qualitative superposition Enrico W Colera Hewlett-Packard Laboratories, Ftlton Road, Stoke Gtfford, Bristol, BS12 6QZ, UK

Received July 1989 Rewsed November 1991

Abstract

Colera, E W , Quahtatlve superposltlOn, Artlfioal Intelhgence 56 (1992) 171-196 Results are presented generahslng superposltlOn to nonlinear systems by using qualitative differential equations These are apphed to allow the composition and decomposmon of qualitative histories Histories record the quahtatlve changes in a system over time, and they can be automatically generated by quahtatwe s~mulatlon The quahtatwe superposmon of such histories is shown to be identical to the qualitative simulation of interactions within linear systems, and many nonlinear systems The result of adding two histories is a partml envlslonment for the system, and the recreation of the interaction history is a path traversal of the envlstonment space The techmque is useful when a reasoning system needs to decompose an interaction history to identify the state of each contributing history, and when examples of a system's behav~ours exist as histories but no model is available Formahslng the hmlts to quahtatwe superposltlon also has implications for other forms of inference such as the resolution of mulhple causal references through a single parameter

1. Introduction Histories record the quahtatlve changes m a system over time They were first p r o p o s e d b y H a y e s [9] as an i m p r o v e m e n t on t h e s i t u a t i o n a l calculus for d e s c n b l n g a c t i o n s a n d c h a n g e A history was " a c o n n e c t e d p i e c e o f s p a c e t~me" w h i c h r e c o r d e d e v e n t s w l t h m a r e s t r i c t e d spatial e x t e n t T h e p a t h a ball t a k e s t h r o u g h t h e air o r t h e r e s p o n s e a p h y s i o l o g i c a l s y s t e m m a k e s to a d i s t u r b a n c e can b e r e c o r d e d with histories H a y e s n o t e d t h a t a h i s t o r y c o u l d b e t h o u g h t o f as t h e e x t e n s i o n o r o c c u r r e n c e o f a p h y s i c a l p r o c e s s This r e l a t i o n s h i p g a m e d fuller e x p r e s s i o n m F o r b u s ' q u a l i t a t i v e p r o c e s s t h e o r y [6], in which physical s y s t e m s w e r e d e s c r i b e d in t e r m s o f t h e p r o c e s s e s t h a t o c c u r r e d within t h e m Q P t h e o r y p r o p o s e d a m e c h a m s m for t h e a u t o m a t i c g e n e r a t i o n o f histories f r o m such p r o c e s s d e s c r l p Correspondence to E W Colera, Hewlett-Packard Laboratones, Fllton Road, Stoke Glfford, Bnstol, BS12 6QZ, UK Telephone (0272) 799910 Fax (0272) 790554 E-mall ewc@hplb hpl hp com

0004-3702/92/$05 00 (~) 1 9 9 2 - Elsevier Science Pubhshers B V All rights reserved

nons Smc~ then the generation ot histories trom system descriptions has bccn mole tormallx tdennfied as quahtan~c slmulanon, ot which Kmpers QSIM [1(t l is a well-known example Hlstolms m QSIM become solutums Io models ~omposed ol quahtatlvc dlttcrentml constraints q h c lelattonshlp between histories and the underlying system model that produces them is leasonably ~vcll undmstood Much less is known about hm~ h~stones r d a t e to one another and how a rt.asonlng system could use such relanonshlps to make useful mlmences In Haves Mamtesto he noted that histories could relate m various ~ays qhc~ could be adlaccnt, both spatmll~ and temporally or thc'~ could bc hybrid m the sense that histories might interact When t~,o alrL~att colhd~ in mid a n , \~e can think o~ each ,is hawng its own history and at tht. moment ol nnpact d ile~ composite hlstOl\ torm~ q he t ~ m c t p h ' o I ~upetpos~tt(m is a well-known mathen~at~cal p~opc~t\ ot hnear s~stems [131 It allows an\ two system states to be added to prodm.t, a ne\~ statc description which ~s qfll conslqent with that system While ~ep~csentm g a pov, ertul tormal techmquc superposmon ~s hm~ted because ~t m general ~s mapphcable to nonhnear sxstems It will bt_ shown below that b~, i c l a \ m g sxstem descriptions through the use ot quahtatwe mathematical representanons superposmon ~.an tn lact be apphed to a ~ ~de range ot nonhnear svsten> Th~s result allox~s quahtatt~ c dcscnptums ot sx stem beha~ours--h~sto!~t.s--to be added quahtan~elx to reproduce behavlomS consistent with the lnlcractton ot thost, hlstont.s Haves [91 saw h~stones as a basl~_ontological p n m m v e but ~ork smt.c then has locussed largely on thc mcchamsms b~, wtnch we can generate, histories H~stones have thus become a bx-produet ol mlerence p~ocedurt.s such as qua]ltat~xc Mnlu]atlOll Quahtat~xc superposltlon allows histories to ont_e ag,un bc considered an nnportant rt.presentatlonal torm m their own ngh~

1 1

Motivation

There ,ire several situations in ~ hJch the superposltlon oI individual tnstoncs to determine interactions is advantageous •

Ouahtatl~e simulation allows th~ merall bchavlour ot a s\,stem ~ t h interacting e~ents to be determined The snnulanon ot mteracnons hovve~er loses m l o r m a n o n We no longer kno~ tht. individual state ol each lndpvldual event within the interaction lust the ettects ot the union through the compound history This becomes a problem when we want to know what individual processes art. up to during the mteracnon Such informanon is necessar~ for example m process monitoring apphcanons If thc task is to monitor and trcat a patient with multiple diseases the state oI each mdwldual disease may need to be tracked within the merall observed behavlour and its mBehattoupal

de~ompO~lttOlt

Quahtattve superposmon

173

dlvldual response to therapy determined [2] Such state information is inaccessible through normal quahtatlve simulation • Behavtoural composttton If we can determine the behavlour of interactions within a system from the participating individual histories, then we only need to generate the individual histories to reason about Interactions Intuitively, this seems to be a simpler method of "naive" reasoning about interactions in physical systems than reasoning from first principles with a model When using histories as the knowledge primitive, a reasoner can also use extra information to constrain the outcome of a situation If, for example, a bird collides wlth an aircraft, we can note that the mass of the aircraft completely dominates that of the bird Of all the potential paths the two might take after the collision, we need only consider the history in which the aircraft's path is unchanged A full qualitative simulation of the interaction system is avoided, all the inference occurring with the individual histories • Reasomng wtthout models One important consequence of using histories as a representational primitive is that in some circumstances one can make strong conclusions about systems that are as yet unmodelled For example, empirical information about typical behaviours of a system may be available, but the system may be insufficiently characterised to have a useful qualitative model Nevertheless quahtatlve superposltlon allows predictions to be made about the interactions of multiple behavlours from empirically identified single behavlours One consequence of the lack of a qualitative model, as will be shown later, is that such predictions may be defeasible

2. Reasoning with histories: an example Consider the monitoring problem posed in the previous section. A physical system has a number of events occurring concurrently within it, and we wish to monitor the behavlour of each lndwldual event along with the overall behavlour The behavlour of fluid flowing within a U-tube wdl serve as an example A quahtatlve model of a U-tube system is shown in Fig 1 The model Is taken from [10] and is composed of a number of qualitative constraints between observable parameters (see Section 3) The U-tube is modelled as two fluid-filled tanks (arm A and B) connected by a pipe We model the fired within the system with parameters for fluid levels, pressures and flows These parameters are related to each other through qualitative mathematical constraints like addition, or statements of functional monotonicity Qualitative simulations using QSIM on the U-tube model following single increments of fired being added to either arm A or B of the U-tube are presented diagrammatically in Fig 2 Prior to the fluid increments, the system

level(A)

presiure(A) Z / ~

level(B)

presiure(B)

4

k~g l ht. QSIM[l-tubLmodaltakenfromkmper~[10] is q m e s c e n t with e q u a l positive fluid levels m each arm r e c o r d e d ~ l t h the l a b e l s a ( 0 ) a n d b ( 0 ) r e s p e c t i v e l y l o l n m a l l s e the s i m u l a t i o n tor an i n c r e m e n t to a r m A we m a k e the fluid lexel in a r m A t a k e a value g r e a t e r t h a n a((}) ( l a b e l l e d a ( l ) m Fig 2) T h e s i m u l a t i o n then d i s c o v e r s t h r e e distract q u a h taUve states as a flow ol fluid occurs betx~een arm A a n d B a n d a new q m e s c e n t state is r e a c h e d with b o t h fluid levels stcad'v at new xalues a ( 2 ) a n d h( 1 ) b o t h h i g h e r t h a n p r i o r to the fluid i n c r e m e n t T h e label a ( 2 ) t o t the level m a r m A lies s o m e w h e r e bctxveen the level b e l o r e a n d a l t e r l m t l a h s a t l o n ~I he s i m u l a t i o n o u t p u t l o r a single i n c r e m e n t to a i m B m i r r o r s that l o r a r m A It Q S I M is m l t l a h s e d to s i m u l a t e the e t t e c t s ot the two fluid i n c r e m e n t s s l m u l t a n e o u s l > t h r e e s e p a r a t e histories are g e n e r a t e d with quite dl~terenl tmal s t a t e d e s c r l p t u m s (see Fig 3) Fhe histories c o r r e s p o n d to the s i t u a t i o n s in w h i c h the a m o u n t ot flmd a d d e d to a r m A is e q u a l to, g r e a t e r t h a n , a n d less t h a n t h a t a d d e d to a r m B T h e relative height ot the initial fluid levels, a n d the d i r e c t i o n ot the net fired flow b e t w e e n the a r m s is d i f f e r e n t in each o! the t h r e e b e h a v l o u r s In the hrst hlstor.x the fluid i n c r e m e n t s are ot e q u a l size, so that no n e t flow oct.urs at this lcxel o! m o d e l r e s o l u t i o n In the s e c o n d history the. flow rate a n d p r e s s u r e d l l f e r e n c e i n d i c a t e that a net shift ot fluid occurs t r o m a r m B to a r m A i n d i c a t i n g that the a m o u n t a d d e d to a r m B was the g r e a t e s t T h e r e v e r s e s c e n a r i o is r e p r e s e n t e d in the third h~stor~ F r o m the p o i n t of w e w ol a r e a s o n i n g p r o g r a m that wishes to t r a c k b o t h c o n t r i b u t i n g e v e n t s , t h e s e t h r e e i n t e r a c t i o n histories are m a d e q u a t t . T h r e e p r o b l e m s are a p p a r e n t

Quahtattve superpo~ttton

levA levB PrA PrB PrDif

t(0)

FIwR

<01oo,dec> <0/**.dec>

175

levA

I

levB PrA


PrS

[ <-**/0, mc> [

PrDff FIwR

i

A1

t(0)/t(l)

levA levB PrA PrB PrDIf FIwR

<0/~, dec> <01.*. dec>

B1

levA levB IrA PrB PrDtf FIwR

<-~t~, mc> <-ootD, mc>

B2

A2

t(l)

levA levB PrA PrB PrDff

<0, std>

levA levB IrA PrB PrDff

<0, std>

FiwR

<0, std>

FIwR

<0, sld>

A3

I n c r e m e n t to a r m A

B3

I n c r e m e n t to a r m B

Fig 2 Histories for fluid increments to arms A and B of the U-tube

• the loss of information about each individual event within the system description, • the ambiguity of the possible o u t c o m e s - - t h e r e is no clear way of informing Q S I M which of the outcomes is preferred, • inltlahsing a function prior to simulation is ambiguous if it is perturbed by both events and the lnitiahsations conflict W h a t is required is a way of mapping the states of the individual histories in Fig 2 to an interaction state in the histories in Fig 3 In fact such a mapping is possible As foreshadowed in the introduction, it corresponds to the qualitative addition or superposltion of the state descriptions taken from the individual histories to generate a state in the interaction history Examining the rlghtmost interaction history in Fig 3, the state additions that produced it are AIB1--+ A2B2---> A3B3 ,

I76

ltl

kvA levB

t(0)

P~A PrB PrDff FlwR


sld> qd> sld> ~ld>

kvA levB PrA PrB PrDII FIwR

AIBI

< ~/0 me> < ~/0 me>

levA levB PrA PrB PrDff <01~ d~c> FIwR ~0/~ dec>

AIBI

k ~~ le'~B PrA PrB PrDff FIv, R

t(0)/t(l )

(recta

~pb(O)/~ dec> < ~/0 me> < ~ 0 me-

A1BI

levA levB PrA PrB PrDfl <0/~ dec> FIwR <0/~ dec>

42B2

k "*A [evB PrA PrB PrDII FIv,R

t(l)

<0 std> <9 ~td>

&2B2

levA levB PrA PrB PrDfl FI~R

A3B~ k~g


std> std> std> std> std> st&~ A~B3

H I , , t o H ~ ~, [ o r a n u l c i ~ . l n L n t t o b o t h a r m ~, ot t i l l [ I - t u b , _

1 e adding state A1 from the increment to arm A hlstor~ to state B1 lrom the increment to arm B history can produce state A IB1 m the interaction hlstor'~ The three dlihcultles presented pre~lously ~hen simulating the progression ot an~ two interacting histories ha~e now been overcome The first ddhcut D was the mablhty to track the effects ol individual histories within the concurrent history This mtormatlon is now made exphclt The second problern that ot amblgmtv ot outcome, can be resolved bv the use ol m f o r m a h o n about the relah~e magmtude ot the influence ol each history on the interaction system during ,,uperpo~mon The process ot state addmon can be biased to ta~our solunons consistent w~th one hlstor) dominating the other Pre~|ouq~ there was no way ot expressing such m t o r m a n o n in the standard QS1M tormahsm Final b , the third problem ot conflicting function mltlahsattons can also be re,~olved through the use of relative magmtude m l o r m a h o n 2 1

Overflew

Given that there is some motivation tor reasoning with individual histories as a complementar~ techmquc to lull quahtatl~e smmlahon, it remains to be determined under what condmons this might be valid The QSIM algorithm

Quahtattvesuperposttton

177

and representation will be used as the exemplar system for the generation of histories throughout the rest of this paper There remains some controversy among researchers working in the area about whether a process ontology like QP theory, or a device ontology [3, 5] are appropriate representations for qualitative reasoning QSIM makes no assumption about the origin of its qualitative constraint models, and so offers a low-level description language for qualitative systems that could fit into either the process or device ontology [8] The following sections will examine the superposltlOn process m more detail First, relevant portions of the QSIM formalism will be presented Next the mathematical basis for quahtative superposltion using the QSIM representation will be presented, including uonllnear systems An algorithm for generating composite histories after quahtatlve superposltlOn is presented next The paper concludes with a short discussion on reasoning with unmodelled behavlours, the role of the superposltion in causal reasoning, and future directions for this work

3. Qualitative simulation Since QSIM is used as the reference simulation system, it will be necessary to briefly restate some definitions from Kulpers' original work [10] Within the QSIM formalism, a system model consists of a number of parameters linked by constraint relationships QSIM generates a state description for the system consistent with given initial conditions and operating ranges for these parameters, and generates all the valid state descriptions that might follow A sequence of such states forms a history for the system being modelled A summary of the QSIM definitions relevant to the results on superposltlon now follows Definition 3.1 (Parameter) A physical system is characterlsed by a set of real-valued parameters f [a, b]--~ R* which vary continuously over time, are continuously dlfferentiable, and have finitely many critical points in any bounded interval Such parameters are called reasonable functtons Definition 3.2 (Landmark) Every parameter is associated with a finite set of landmark values which must include 0, f(a), f(b), and the values off(t) at each of its critical points Landmark values are considered to be the only interesting values from which a quahtat~ve state description need be drawn Definition 3.3 (Ttme potnt) Distinguished time points are those points where something important happens to a parameter, such as passing a landmark value or reaching a limit Reasonable functions have a finite set of distinguished time points and landmarks values

/ ~, ((mra

17~

Defimtlon 3.4 (Quahtanve slate) The qualitative state ol a p a r a m e t e r consists ot its ordinal rel,mon within the landmark ~alues and its direction ot change It

l~-

-:l/,arethelandmarkvaluesoll

[a,h]--,~',thentoran) tC[a,b], the quahtatlve state ot /' at t ~s QS,( 1, t) and ~s a pair (qval, qdtr} dehned as tollow~

qval=

l, (l~ I,+~),

m~ qdt, = { dt~t~ "

11 f ( t ) - l I 11 f ( t ) ~ ( l j , l + ~ )

fl /'(t) ,t f'(t, it t ' ( t )

0

() ~}

Definition 3.5 (Quahtattve hehavuml) The qualitative behavlour ot j on [a, b] ~s the sequence of quahtatxve states ot f,

QA(l,t,~),QS(l,t,,

t~) QS( t t,),

Q~(I t,, , t,,) Q3(! t,)

alternating between quahtauve states at distinguished time points and quahtatlve states on intervals between d~stlngulshed t~me points

Definitmn 3.6 (System) A system is a set ot reasonable tunchons that are related b) a set ot quahtatlve differential equations and the behav~our of the system ~s the union ol the behawours ot its tunct~ons Every quahtatlve state has a qualitative description, and that description tan change at dlstlngmshed t~me points and remains constant on thc open intervals between them The quahtat~ve state ot a system F of m tunchons is the m-tuple ol individual states

(?S(F t , ) - [ L ) , S ( / ,

t)

(23( /,,,, I. II

The qualitative b e h a w o u r ot F is the sequence ol states

Q,S(F t,~) Q3(F t, )

, QS(F, t,)

Defimtion 3.7 ((onstramt) A system model mvoNes detailing tunctlonal relationships through a n u m b e r ol qualitative tunctlonal constraints Constraints may be two- or three-placed predicates, and they restrict the quahtatlve values that may be assigned to functions The Q S I M constraints are • A D D ( J, g, tl) lff f(t) + g(t) = h(t), • M U L T ( J , g, h) Ifl f(t) × g(t) = h(t), • M I N U S ( J , g) fff J ( t ) = - g ( t ) , • D E R I V ( J, g) ~ft f ' ( t ) = g(t), • M+( f, g) Ill ] ( t ) = H(g(t)), H(~) IS strictly monotonically increasing, • M ( t, g) fit l(t)= H(g(t)) H(x) is strictly monotonically decreasing

Quahtative superposttton

179

We can consider these constraints as quahtatlve abstractions of ordinary dffferentml equations (ODEs) We can decompose any O D E into a set of simultaneous first-order equations which can be replaced by a qualitative constraint Thus a system model can be considered to be a set of quahtattve differential equations (QDEs) Any behavlour that satisfies an O D E must satisfy the corresponding Q D E Since many possible /unctions can map onto the same qualitative constraint however, a given Q D E may be the abstraction of numerous O D E s

4. Qualitative superposition Just as a qualitative model can be an abstraction of ordinary differential equations, a history can be regarded as an abstraction of the solution to such a set of equations Two histories can add to reproduce an interaction h~story precisely when two qualitative solutions for a system can be added to produce another solution to a system ~ It is a standard result from the study of ordinary differential equations that for a given hnear system of differential equations, the sum of any two solutions to the system is also a solution This superposttion property follows from the existence of hnearly mdependent solutions (Appendix A 1) Thus whenever two histories are combined and the histories are from a hnear system, a true solution to that system results However, th~s is simply not true in general for nonlinear systems Yet it is precisely nonlinear systems that are of interest since most of the physical systems that are of importance &splay nonlinear behaviour One way of extendmg the property of superposablhty to nonlinear systems is to weaken our requirements In particular if we require superposmon to only successfully operate on the quahtatwe behavlours of functions rather than their strict numeric values, we may still be able to derive useful inferences with it The approach taken below will be to identify conditions in which qualitative solutions from nonhnear systems behave as if the system they came from was linear Superposltion has varying degrees of vahdity, depending on whether the system it 1s applied to is linear or nonlinear, whether quantitative or quahtatlve solutions are desired, and on the nature of the nonlinearity Four validity classes can be identified

1 In general, when two h~stones interact, this need not occur over all parameters m each history This is especially the case when two subsystems c o m m u m c a t e with each other through a small subset of their parameters T h e Interaction occurs only through the shared parameters, and this is specifically the area we are interested in A slmphfymg assumption for the remainder of this &scusslon will be that non-shared parameters can be treated as a "black box" when dealing with interacting subsystems and their interacting histories We will only be interested in the inputs from the non-shared parameters into the shared system

180

1 It

( ou'~a

•

T~pe 1 T h e Principle ot S u p e r p o s m o n apphes uniformly tot the quantitative values ot all lunctlons m a system All linear systems lall into this class • Type 2 S u p e r p o s m o n apphes lor quahtatlve ~alues ot all lunctlons m a system Functional r e l a n o n s are preserved between the \ a l u e s p r o d u c e d by s u p e r p o s m o n • l ~ p e 3 Functional relations ~.annot be maintained tollowmg quahtatlve superposltlOn but the resulting solution d e m o n s t r a t e s the correct quahtatlve b e h a v l o u r This means that the s l g n o l a l u n c t l o n a n d l t s d e r l x a t l x c can be predicted but not its ordinal position in a l a n d m a r k list • T~pe 4 Quahtatlve s u p c r p o s m o n i v, not s u p p o r t e d 4 1 (;enetal a p p t o a d t In the general case a ph)smal system ma~ be represented bx ,in n t h - o r d e r dlfferennal equation It was noted above that such an equation can be reduced to a series ot hrst-order equations, whmh can each be m a p p e d o n t o Q S I M constraints ( A p p e n d i x A 2) The p r o b l e m no,a reduces to l d e n n t y m g c o n & n o n s u n d e r which ;~e can add s o l u n o u s tor each of the quahtatlvc constraints and derive solutions that ale still xalld for these constraints We shall call a constraint that sansfies such a c o n d m o n a reasonable constraint dnd d e h n c It thus

Definition 4.1. Let ( be a qualltaUve constraint ot the torm (_(a, I~) where a and b are reasonable t u n c h o n s I1 a I (l~ ¢1,, are solutions tor a and bj h~, h,, arc solutions lor h such thal ( (a~, h I ) C ( a , b:), , C(a,,, b,,) are true, then it tor an} two constraint solutions t and ~ where ~ [ l l ;11 and ~ , ~ [ 0 ;l I C ( a , + a b, + h ) Is true then ( is a teasonahle t on ~tt atnt A smular definition can be t o r m u l a t e d lor three-placed pre&cates Defimtmn 4.2. It F is a system ol reasonable constraints p r o d u c e d with k is a reasonable htstotx

4 2

then any behaxlour

5uperposttton wuh the Q S I M {onst~amts

It can tii~iall~ bc s h o ~ n that the QS1M constramt~ M I N U S A D D and D E R I V are reasonable since these constraints capture linear beha~lour Solutions will thus be of type 1 (or 2 1l so desired) Nonlinear behavlours lie in the m o n o t o n i c and multiplication constraints We can d e m o n s t r a t e that the M ~ and M constraints support type-2 qualitative superposltlon This is not the In C)%INI corrc',ptmdmg ~aluc,~ t.ammt bt_ ,,upportt_d lot t~,pt.-~ ~,olutum,,

Quahtattve superposmon

181

case for the MULT constraint However, making an assumpnon about the relative magmtude of the two solutions being combined is a sufficient condmon for the MULT constraint to allow type-3 solunons to be produced The following proofs demonstrate the degree of superposltlOn possible for each of the MINUS, ADD, DERIV, M +, M-, and MULT constraints The constraint defimtlons gwen here are taken directly from [10]

4 2 1 The hnear constramts Definition 4.3. MINUS(f, g) is a two-placed predicate on reasonable functions

f,g [a, b]---~* which holds lff f ( t ) = - g ( t ) for every t ~ [a, b] Proposition 4.4. If MINUS(fl, gl) MINUS(f1 +f2, g~ + g2) holds

and MINUS(fz, g2) are true, then

Proof. Gwen fl = - g l and f2 = -g2, then fl + fz = --gl -- g2 -- ( gl + g2)

[]

Definition 4.5. A D D ( f , g, h) is a three-placed pre&cate on reasonable functions f,g,h [a, b]---~ ~* which holds lff f(t) + g(t) = h(t) for every t E [a, b] Proposition 4.6. If ADD(f~, gl, h~) and ADD(fE, g2, h2), then ADD(fl + fz, gt + g2, hi + h2) holds Proof. This result clearly follows from the definmon of A D D

[]

Definition 4.7. DERIV(f, g) is a two-placed predicate on reasonable functions f , g [a, b ] ~ * which holds lfff'(t)=g(t) for every tE[a, b] Proposition 4.8. If DERIV(f~, g~) and D E R I V ( f 2, g2) are true, then DERIV(f~ + re, g~ + g2) holds Proof. Gwen g~ = dfJdt and gz = dfz/dt, then d dfl df2 at (fl + f 2 ) = ~ + d--i = (gl +

g2)

[]

4 2 2 The nonhnear constramts Definition

4.9.

M +

is

a

two-placed predicate

on reasonable

functions

f , g [a, b]---~* M + is true l f f f ( t ) = H ( g ( t ) ) for all tE[a,b], where H i s a

182

l !,~ ( o t t t a

function w~th domain g([a. b]) and range /([a hi) H ' ( t ) > 0 for all ~ m the interior ot the domain

dflferennable

and ~ t h

The essential property captured m a monotonic increasing rclanonsh~p between two luncnons is that they reach crmcal points at identical dlstmgmshed hme points and that m the mter~emng regions they' share the same direction ol change [101, r e t ' , ~ ' -0 It is th~s lunchonal relahonsh~p that need to be preserved atter superpos~hon Proposition 4.10. (;wen M ' ( J , g) and two solutton patr; ( 1~, gt) and ( L e:) we tan show that M ' ( tl + [: ,~1 + g2 ) also holds Tiros we seeh to demonstrate that a Juncttonal relatum3htp eusts bemeen ( [i gl) attd ( f l , g~) suth that ( 1 ) ( ¢ '~+ l '~)/ ( g'l + g~) > l) between ~i ttt~ al points (whet e g'~ + g~ t s defined ) (2) [i + t'~ attd gl + g~ teach ~ttttcal potnts at the same tmu" Proof.

( 1 ) Behavtout between ctttt~al points Given that L - H(g~) and l~ we note that [I + [ : = H ( g t )

H( g_ )

+H(g:),

['~ + 1~ = H ' ( g , ) x g'~ ~ H ' ( g : ) x g~ l'~ +t~

H'(gl)xg

I

H'(g~)Xg'~

t

r

Let A -

H'(g~) ~ gl

and

B-

H ' ( g : ) < g'~

We now evaluate the sign ot the expression ( 11 t- [':)/( g'l + g':) tor ~alue~ o! gl and g':

Case l Let g'~-O Then A = 0 and B = H ' ( g 2) which vve know tram Defimtlon 4 9 is always posmve Case 2 Let g'~=0 Similarly B = 0 and 4 = H ' ( g ~ ) which is also always posmve Case 3 Let g'l g ' ; > 0 Tnvmlly both A and B are posmvc Ca~e 4 Letg'~, g': 0 Let t and ] take the values 1 or 2 In both c a s e s A + B is poslnve, when

(1) Ig;l>lg;I and (2) I H ' ( g , ) l > l H ' ( g , ) [

Quahtatwe superposttton

183

Th,s second condltmn ,s equivalent to stipulating that If;/g,'l > If;/g~l, or noting that condition (1) must also hold, simply that If;I > If l These two conditions, Ig;I > Ig~L and If;I > IfSI, together Imply that the effect of one function's values dominates the other's during the superposltion and collectively form the relative magmtude constraint Cases 1-5 are summarlsed m Table 1 It may be possible to specify further conditions for superposltion in Case 5 by closer examination of the function H For example if we know it to be linear, then the magmtude constraint may be dropped If it is weakly linear such that Ig;I > but that IH'(g,)l is only mImmally less than IH'(g/)l, then the expression H ' ( g , ) x gl may still evaluate to be greater than H ' ( g ~ ) × g~ and satisfy the sign of derivative requirement

IgSI,

(2) Behavtour at crtttcal pomts We need to show that f~ + f~ and g~ + g2' reach zero at the same time There are two cases at which g~ + g2 = 0 that need to be examined

Case 1 Both original functions reach critical points at the same time By defimtlon, when gl = 0 then f~ = 0 Thus when f~ and f~ are zero, g'l + g~ = 0 and the expression (f[ +f2)/(g[ + g2) is defined and equal to 0, implying that both f~ + f~ and g~ + g" reach critical points when the original functions reach a crmcal point Case 2 When Ig l = Ig l but are opposite in sign, then g[ + g2 = 0 However f~ + f~ may be nonzero, making (f~ + f ' ) / ( g ~ + g~) undefined In this case one superposed function may reach a critical point without the other doing so This case is avoided by the stipulation m Case 5 of the first part of this proof that a relative magmtude constraint must be enforceable for superposmon to be possible when functions being added are of opposite sign A similar proof can be constructed for the M- constraint

Table 1 Summary of type-2 quahtatlve superposttmn results using the monotonic increasing constraint, given M+(f~ +f2, g~ + g2), RMC ln&cates that the relahve magmtude constraint apphes

g" g[

+

o

-

+

~

V'

RMC

o

V

V

V

-

RMC

x/

v'

[]

184

1 ~'~ ( r ) l t / a

J, g h ) i', a three-placed predicate on reasonable tunc[a, hi--+~" which holds ill ]'(t) * 7 ( t ) = h ( t ) lor cverx t ~ l a b]

Definition 4.11 M U L T (

tions f g,h

In the general case, it f, x g, = h, and l, \ g, = h, then ( l, + f,) ~" ( v ~,) (h, + h r) H o w e v e r , v,c are interested in identll2cing situations in which the expression is still quahtatl~cl 3 truc i e It, + h, displays the saint quahtatl~c b e h a v i o u r as ( [, + l, ) "~ (g, + k',) to p r o d u c e a type-3 prediction This reduces to two s u b p r o b l e m s • W h e n is the sign ot the t u n c u o n value p r c s c r \ e d ~ • W h e n is the sign ol the derivative p r e s e r v e d ~ Proposition 4 12. 1I M U L T ( I, g,

Ill)

alld M U L F ( l, g., ]l,) ate title attd we

can sl~e(tf~ ( 1 ) the #elatlie ##lil<~lll[lld(" cO#l%Dill#l[

/I//t)l

- I t~_(,)1

I/;(,)i

//i
->1<<(,)1

I,
1<4(,)1)

.,.z

O#

II/,It/I ~//~/~/I I,<(t)l <- I<<(~)1.

I/;(t)l

~1/k(t)l

~Hld

I.~.i(,)1 - I~'(t)l)

()t

(2) a similar eqmvalence constraint

(I I,(,)1- I I~(,)1 I.<(t)l = t,<(ol fol a l l t C [ a , b

I/;~,)1- I tA(t)l

(llld

I,~;(,)1- I <<~,)1)

then

slgn(h

+ h, ) -- sign(( l, -~ L d

(gl + g - i ) d

))

Proof. ( o n s l d e r the relative m a g n i t u d e constraint (1)

real n u m b e r s a and h that it

I.I ~ lt, I

sign((li + f2) (gl +
Note the result for two then sign(a + b) = sign(a)

slgn(fl

gi)

- slgn(Jlgl + Mg:)

-slgn(h~

~ h.)

Quahtattve superposltton

185

Further, let

x=(L+L)

(gl + g2),

then x'=(]'l+L)'

(g~+gz)+(fi+f2)

(gl+g2)'

= ( f l + f ~ ) (g~ + g2) + (f~ +f2) (g~ + g') Thus slgn(x')=slgn(f~

g, + f l

g[)

Let

y = h, + h2 = ( L g , +f2gz), then

Y' = f;gl + Lg~ + f~_g2 + L g " Thus sign(y') = slgn(f~ gl + fl g~) = slgn(x') A similar argument can be produced for the equivalence constraint (2)

[]

5. Generating composite behaviours Having demonstrated that qualitative superposltlOn is valid for most systems expressible in the QSIM representation, ~t is now necessary to demonstrate how this property can be harnessed to generate interaction histories Given two reasonable histories originating from the same system, how do we actually generate a composite behawour to represent their interaction 9 There are two fundamental steps to this process (1) Superposmon Adding qualitative states to produce all possible composite states, (2) Assembly Concatenating composite states into a legal behavlour for the system The soundness of addition for reasonable histories means that any two such histories derived from the same set of QDEs can be used An order-preserving addition of states needs to be performed, 1 e a state from one history is added to al! the states in the other history with which it might possibly Interact However, addition is an underconstrained process Other solutions are also produced, along with the correct one, because of the inherent ambiguity of

quahtatl~e addition Adding QS,(tl /~ tnc) to Q)~,;,(ll': de~) produces cnhcl C)5.~/,(0/7-.m~) C)~,,;,(I)'~ stdl o~ C)~,, :,((~ ~ ,h'~) Fmthc~ sucl~ -olut~ons could be assembled into mdehmtelv long behav~oms L

()S

:,((I),'~ ;m },t,,

()S,, /,(
()S,, :,((ll,z de~) :,, i) QS,_:,((()/':-.z;zc> t

)

Q~,, /,((11 :- s;d) t,, ~ t, ,) QS

:,l(()'~ std) t .~';,,

This problem is called "chatter [ 111 and results from the inherent amblgmt~ ot quahtatwe simulation as ~ell as lrom ploblems wnh the Iocaht\ ol translhon value selection m O S I M These problems can bc controlled b~ • filtering c o m p o s n e states that arc not ~ahd s~,stem beha~lours during the quahtatwe a d d m o n • entorclng behaxloural contmmtv ~ h e n assembhng sum states into neu histories Both ot these techniques wdl no~ be explored m more detail

5 1 3upetposztu)tt

(tearing att(t [zlwptttg (omt?ostte states

Although adding two quahtatl~L states taken horn two histories will produce unwanted states along with the real ones there are at least two ways that such states can be e h m m a t e d (1) Usmg the ortgmal ~sletn ~on~ttalnts The sum states can be hltered through the system model that produced the original h~stones ~e a c o m p o s n e behavlour must be legal lor the system that produces ~t (2) Making an assumptton about relame magmtudes k n o w i n g the magmtude ot the etlect each hlstor 3 has on the interaction allows a decision about which history dominates during the a d d m o n to be made Whde thns is reqmred to establish reasonableness tor some nonhnear s~stems, ~t also chromates spurious solutions during state a d d m o n Rather than generating all possible states derivable from quahtatl~c a d d m o n and then hltermg them it is easier to hltel them during the process ol quahtatlve a d d m o n In etfect, a single-pass quahtatwe slmulanon is p e r f o r m e d using the system model along with a d d m o n constraints between the behawours ot mterest Further, rather than performing a lull p e r m u t a t m n o| state a d d m o n s only those states that preserve lunctlonal or temporal continuity need to be added (see Section 5 2) In p a m c u l a r transients trom contributing histories cannot

Quahtattve superposttton

187

exist more than momentarily Such transients, when they occur, are present as initial states of the histories being added We avoid adding Initial states with transients to states from the other history other than its initial state Hence states A1 and B1 in Fig 2 can only be assembled into composite state A I B 1 A state A I B 2 would imply that a history of A1B1--~ A I B 2 was possible, implying incorrectly that the A1 transient could be extended over the two quahtatwe states, a point and an interval The state generation algorithm first selects the states to be added, and then performs the addition by simulation Once two states have been selected for addition, the superposltlOn by qualitative simulation occurs in three stages Assume a system of function~ F, a set of constraints C, and two reasonable histories H 1 and H 2 from F and C For each pair of quahtatlve states to be added we perform the following steps

Step 1 Compostte model assembly (a) For each function f E F create three copies labelled fl, f2, and fsum Call the system of new functions Fsum (b) For each function triplet so created, generate an A D D constraint A D D ( f , , f2, Lum) (c) For each constraint c E C generate a copy cs. m between the sum functions Fsu m Call the new A D D constraints and the copy of C together Csum (d) Landmark values, domain restrictions and corresponding values for each f E F are transferred for each copy function fsum ~ Fsum Step 2 Compostte model mtttahsatton For each function fl E Fsum lnltlahse the function value to its value in H 1 Repeat the procedure for each f2 E Fsum using values in H 2 Step 3 Quahtattve slmulatton on the compostte model Using QSIM, perform a single iteration of the algorithm on the composite model F~um and C,u m to find legal values for each f~.m and assemble these values into legal state descriptions for F~um Each coherent set of value assignments to the members of Fs. m is a solution to the addition During the quahtatlve simulation on the composite system model, a restricted table of possible solutions for the A D D constraint can be used when one state in the addition is known a priori to dominate the addition (Tables 2 and 3) Such knowledge is needed when superposltlOn requires a relative magnitude constraint, but can also be used to assist in reducing the possible states derived during the simulation The result of this process for two states taken from two reasonable histories is all the possible interaction states derivable for those two states The next step in the superpositlon process is the linking of these states into complete histories representing Interaction behaviours

lablc 2 l , m d m , u k ,iddlholl t,tblc V~llh rclatlx,_ m a g m t u d e as~umption ](.Jl IP]

{t

5 2

Path assembly

{I

{~

II

etfforcmg behmtoutal ~ontmtttt~

The techmque oI em lStomng generates all the possible state descriptions tor a %'stem model, based on a set o! background assumpnons An~ speclhc hlstor~ tor the system corresponds mlormallv to a path through that envls~onment It we ha~e two leasonablc h~stones, and do a permuted stat~ addmon between the two, a state space ~s dehned which ~s eqm~alent to or is some subset ol a full enx~stonment The task ol assembhng an interaction tnsto~x trom th~s ne~ state space thus becomes one ot path traxersal A logu o1 o~¢u.en¢e was described by Forbus [7] which presented a tormahsed relanonshlp between h l q o n e s and cmlstonments In pamcular, the task ot selecting a path through the envlslonment tot a pamally created hlstor~ was described In our case. iI we have selected a state from the pamal enwslonment created by h~storx addition, ~e need to determine ,ahlch states are possible next states for the mteracnon history Often addmonal mlormanon about the speed ot complenon of each tustorv and the synchronlctt\ ot transmon to subsequent states ~,tll be a~aflable Such m t o r m a n o n greatb conqralns the search tot a path through the pamal enwslonment A method lot path tra~ersal that allows mtormatlon about the mdl~Mual interacting beha~lours to be unhsed will be presented belm~ In a~.cordancc with Forbus logic ol occurrence we assume here that the histories ~e will generate are brute--that they will terminate at some time m the future We will also assume Forbus termmologx which ~s bnefl'¢ dehned now Defimtion 5.1 (Lnvlsumment) An cnvlslonment t leprescnts all possible quahtatlve states a pamcular system mav take on and all legal ttansmons between them labk Direction ot IDERI\(P) I

change

,tddttum

table

DFRI\(t')

,alth

rclatt,,c

tnagnflud~_ m~_

a',sunlptlon

JI)P R I \ ( O l[

,rid

dec

111~-

I[1~

liB.

dec

,,td ckc

duc

DLRIV(C) I I l'l ~..

',td dec

Quahtattve superposttton

Definition revolving transition transition

189

5.2 (Transttton functton) Transition functions describe all transitions a state s State s, E Befores(s) when the envlsaonment contains a from s, to s State s, E Alters(s) when the envislonment contains a from s to s,

Definition 5.3 (Path) A path is a sequence of states s,, s,+ 1 ~ Alters(s,)

, s n such that

Definition 5.4 (Status of states) Consider a history H involving envlslonment e For every state s EStates(e), exactly one of the following is true posstble(s, H ) , requtred(s, H), excluded(s, H) Simply, a state is posstble ff It may appear in a history, excluded if it never occurs, and requtred If It must appear Forbus introduces mechanisms for handhng cycles, and avoiding impossible cyclic behawours These techniques for path traversal of envlslonments are directly applicable to the state space generated by summing behaviours, and will not be reproduced here

5 2 1. Path creatton U n h k e the envlslonments in Forbus' work, states produced by superposltlOn are not explicitly linked The process of history generation by path traversal must thus Include a step that identifies candidate next states Given a current state current(s) m a h~story H the next state in the path is defined as

Vcurrent(s) 3next(r) ~ r E States(e), -nexcluded(r, H) We need to identify next states that are legal transitions, before we can decide whether they are excluded Possible states are identified using restrictions imposed by temporal continuity, and excluded states by application of rules that ensure functional continuity through the assembled history

5 2 2 Temporal contmutty Next states can be identified with knowledge about the temporal behavlour of the original contributing histories In particular, each sum state carries with it the labels of the distmgmshed time points or intervals taken from its contributing states Next states are identified with reference to that label The search for legal next states can be coded in several rules Assume two reasonable histories/4, and Hj and states drawn from each history s, and sj Let s, have qualitative description QS(f,, t,) and sj have QS(fl, tj), and call the resulting set of summation states S, j Each state s E S, j is then labelled

QS(f, +,, t,, t,)

101)

/ !4

(ot~a

Rule 1 ( T e m p m a l Ptogte~slott) Fol a state s C 5 ,, which is the c m l c n t last state m ,t history H and has q u a h t , m v e state Q,S(/, , t,, t;) ~ c hax~

poss;bh'(H, s ) = { Q S ( I

, t, t,) QS( / , t

QS(I

, ;,.t,,,)

~ t)

(2s~(f, , ;

t

,~;

T h e T e m p o r a l Progression Rule prevents local cycles being generated and e n l o r c e s the m t u m o n that, lor the interaction history to progress to a new quahtatlve state, either or both of the contributing histories must also plogress N o t e that two sequential states m a path ma} be the result of the same state a d d m o n , l e a state with a Q S ( / , , t , t ; ) label can be a next state lor a s t a t c with the same label This reflects the possfl~lhtv that the a d d m o n ot two states can p r o d u c e m o r e than one interaction state T h e "Iemporal Progression Rule can be speclallsed m somc cases to tmhsc additional knowledge a b o u t the relationship between interacting hlstollcs 111 p a m c u l a r l n t o r m a t l o n about svnchromclt~ o! history evolution and speed ol c o m p l e t u m ma~ be utfllsed to constrain the process o! path creation Rule 2 (,Synchronous Progress;on) A s s u m e two leasonable historic,, t t a n d / - t have equal path length For a state s C S , x~hlch Js the current last state m ,~ hlstor} H and has q u a h t a n v e state Q S ( / , , t t.), we haxc

l?os~;ble(H, s) = Q S ( / ,

, ;, i t,~ i)

It we know that the lntelactmg histories wall both progress to a ne~ d l s t m g m s h e d t~me point in a s~nchronous m a n n e r then x~c can exclude asynchronous combinations Rule 3 ( A s y n c h r o n o u s P m g r e s s t o n ) For a state s ~ ~, ;, which ~s the current last state m a history H and has q u a h t a n ~ e state QS(/,+, t, t,) ~ c haxe

posstble(tt.s)=

{QS([,.;

t , + l , t ,) QS(I,

, t

t,,~))

T h e A s y n c h r o n o u s Progression Rule enforces a s y n c h r o n o u s progression ol the contributing histories Rule 4 ( S p e e d o] Progression) A s s u m e two reasonable histories H, and ft, ot equal path length where tot any interval (t,, t,, L),

d u ; a t t o n ( H , , t,, t ~ E) < duratton(H;, t, t, ~) T h e n , for an}' d l s t m g m s h e d tnne point with statL s C S, ; and with quahtatlVe value QS( /, ,, t, t;,), Ia ~

Ii,

Quahtattve superposttton

191

The Speed Rule ensures that the faster of the two contributing histories reaches its final state first In applying the Speed Rule, we are making assumptions about which particular path among several alternatives we take to create our composite history The effect of this rule is not to eliminate possible histories, as the synchronlcity rules do, but to eliminate alternative paths for each history It reduces the size of the search space in the sum envislonment, not the possible histories the envisionment contains

5 2 3 Funcnonal contmutty Once the set of possible next states has been identified, the path creation algorithm chooses only those states that maintain functional continuity For example, two states would not be temporally adjacent in a history if a functional discontinuity exists between them, e g QS((O, mc),t,) and QS((O, dec), t,+l) would not be allowed Further, continuity determines whether adjacent states can have the same description In particular, a transition from an interval to a time point should not have the same qualitative state since distinguished time points represent critical points, but the reverse is allowable [10] Functional continuity applies not just to the history being assembled, but to the histories that contribute to the assembly The process of selecting possible next states ensures that the contributing histories are traversed in a legal temporal sequence Along with the composite history, we need to ensure that states from the contributing histories are not prolonged unnecessarily As indicated in Section 5 1 transients in the histories contributing to the superposltlon can only exist for an instant, and state combinations that would prolong them are filtered during the initial state generation Functional continuity could also be invoked in the process of value recovery As we saw in Section 4 2 2, there are some values for functions in a monotomc relationship which prevent superposltion However, by looking ahead to the next state, it may be that superposltion is once again possible Where this is the case, the non-superposable state might be recreated by interpolation between its predecessor and successor states, relying on the fact that function values must be continuous

6. Qualitative superposition: some consequences A n u m b e r of consequences arise from the results in this paper which both expand the repertoire of qualitative reasoning systems and also impose limits on some types of causal reasoning Two particular issues are explored h e r e - the notion that histories can be used independently of deeper qualitative models and the validity of the assumption that causal influences can be considered linearly independent

lq2

t ~$ ( o . ; .

0 1 Htsto;tes wttkottl models In s o m e c i r c u m s t a n c e s we might w a n t to m a k e p r e d i c t i o n s b a s e d on hlstolleS t h a t w e r e not g e n e r a t e d f r o m a m o d e l This might be the case when e m p i r i c a l I n f o r m a t i o n a b o u t a s y s t e m s b e h d v l o u r IS a v a i l a b l e but t h e r e is lnsutficlent u n d e r s t a n d i n g of its m e c h a n i s m s to c o n s t r u c t a q u a l i t a t i v e m o d e l In such c i r c u m s t a n c e s we do not know w h e t h e r the histories x~e arc m a m p u l a t l n g arise from h n e a r or n o n h n e a r s),,tems or m tact d mdc~.d the h~stones c o m e l r o m the s a m e , v q e m It the b e h a ~ o u r s r e p r e s e n t tault~ b e h d v l o u r s . It m a ) be that a d f l l e r e n t tault m o d e l p r o d u c e d each bt.hdxlOUr D i f f e r e n t m o d e l s clearl~ Imply dffterent sets ol system e q u a t i o n s a n d m such s~tuatlons s u m m i n g s o l u t i o n s Is m e d m n g l e s s A c o n s e q u e n c e ol tht.~c unc m t a m t l e s is that, a l t h o u g h a m e c h a m s m no'a exists to p r o d u c e p r e d i c t i o n s a b o u t m t c l a c t l o n s d~rcctly trom behav~ours such pre&ct~ons must be consldc r c d deleastbh' Recall that a l t h o u g h Q S I M g e n e r a t e s s o m e b e h a v l o u r s that arc not true tor a s y s t e m ~t wilt always g e n e r a t e all possible true b e h a ~ l o t t l s Vv'c n o a ha~e no such g u a r a n t e e F o i a n o n - m o d e l l e d b c h a v k m r to be usclul it s h o u l d bc well-to;reed B) this we m e a n that It d l s p l a ) s ,ill the f e a t u r e s ot a true m o d e l - g e n e r a t e d b e h a v l o u r sut.h as c o n t m m t 3 ol t u n c h o n b e h a x u m r F u r t h e r ~t ~s Ideal it a rclatlx~_ m a g m t u d e a s s u m p t i o n b e t u e c n the s u m m e d b e h d x l o u r s can be m a d e [his is so for t,~o r e a s o n s T h e a b s e n c e o l d m o d e l m e a n s that the a d d m o n ol quahtatlXC states is u n d e r c o n s t r a m e d - - w c do not haxe a m o d e l to h l t c r ou! i l l - b e h a v e d s o l u t i o n s T h e use ol rclat~xc m d g m t u d e mtormat~on s u t h c l c n t l ) c o n s t r a i n s the add~tum p r o c c s s m such c~rcumstanccs S e c o n d l ) x~c s h o u l d m a k e a w o r s t - c a s e a s s u m p t i o n that the u n d e r l w n g system m o d e l is n o n h n e a r Bx a s s u m i n g that o n e h~story d o m i n a t e s tht. s u p e r p o s m o n wc ensure, that x a h d p r e d ~ c t u m s are m a d e 1ol systems that c o n t a i n m u l h p h c a t ~ e const~a|nts bct\,~c t.n t u n c t l o n s

(~ ~

( omhltlltl,g [altsO[ slttle~

C a u s a h t \ rod) be r e p r e s e n t e d a', the t e m p o r a l p i o g r c s s l o n ot states within a %stem Forexample take aCASNET[14]hnkbctweentwostatesmvolvedm the p r o g r e s s i o n ot the eve dmseasc g l a u c o m a L l e x a t c d P r e s s u r e T r a n s m i t t e d to Optu. N e r v e H e a d - - , (t)t)) D e t . r c a s c d B l o o d Suppl~ to O p t i c N e r v e H e a d P a t h s t h r o u g h causal n e t w o r k s c o m p o s e d ot such states are direct a n a l o g u e s ot the qualitdU~e histories d~scussed m this p a p e r It m a y be the ~.ase that the e t t e c t s ol two states eMst c o n c u r r e n t l y , a n d are r e s o l v e d to p r o d u c e a t h i r d m a n e t w o r k - - d p r o c e s s a n a l o g o u s to s u p e r p o s m o n It we do vmw the c o m b i n a t i o n oI statt.s as q u a h t d t w c s u p e r p o s m o n then the results p r e s e n t e d e a r l i e r e s t a b -

Quahtattve superposttton

193

hsh that a number of conditions need to be met prior to combination In particular the states must be (1) Dertved from the same structural model It is meaningless to use superpositIon on states that come from different models of a system-the procedure assumes that they represent solutions to the same set of qualitative differential equations Consequently when states are combined they must belong to the same model States that come, for example, from different fault models of a mechanism cannot be combined (2) Sattsfy condztzons of reasonableness If the system that produced the causal states is modelled with multiphcatlve and monotonic functions, then a condition like the relative magmtude constraint may need to be met to make the addition valid Failing either condition implies that inferences derived from the combination of states are unsound

7. Future work

This paper has presented an alternative qualitative reasoning methodology, utihslng qualitative histories to determine the effects of interactions within a system The similarity between adding histories and path traversal through an envisionment was also identified Several avenues for further development of this technique are apparent

• D t a g n o s t t c monitoring The techniques in this paper have been developed for, and incorporated into a program for the diagnostic monitoring of changes in patient state over time in the medical domain of acid-base disturbances [1, 2] How they may optimally be used in conjunction with standard qualitative simulation remains to be determined Other work with the monitoring of dynamic systems using qualitative simulation will serve as a useful benchmark, e g [4] • Reasonableness condttzons A limitation of the current work is ItS application to nonlinear models incorporating MULT constraints (and to a lesser extent monotonic constraints) where situations arise in which qualitative superposltlOn cannot be performed However, there IS no reason why other Instances where reasonable behavlour is exhibited cannot be found amongst these, and it is the identification of such instances that will enhance the applicability of the behavlour addition methodology • Representing structural faults For the techniques in this paper to work the underlying qualitative model that produced the interacting histories must be the same Yet many abnormal system behavlours are generated by

194

[ It (,n~/a

introducing structural changes into the model One solution lb to rcplacu single taulted constraints ~lth subs\stems that re~eal lnoru cKtaflcd mechanism structures [121 Fhus l,lthel than altering the tault~.~l ~.~m stralnt we can ,it the more detailed level hold parameters con,,tant ol perturb their ,alues, thus reproducing the faulty behaxlour at the higher level This method mav not always be possible and the plactlcalltx, ot pertormlng such transformations need to be more lully explored • ( o m p a r a t t t ' e analvst~ Weld's work on comparative analysis ot ~ s t e m s [15] takes a system behavumr and explains how that beha~lour will altel after perturbation The inethodology is limited when the pcrturb,mon causes a topological change in the system behavnour Therc is ~llmlarnt,, between the determmatlon ot l e l a m c change in Weld s ,aork and det c r m m m g the quahtatlve dltlerence of two behaxlours (Determining the difference ol two behavlours ~s perlormed in an identical manner to adding two behaxlours ) Since there is no lnmmmon on the topolog~ ol behaxlours in the work plesented in this paper It is possible that It llla_~ haxe relexance to comparative anal\sis

Appendnx A This appendix restates results demonstrating that for any linear system a set of linearly independent solutions can be found that allow the sum ot any system solutions to also be a system solution and that any nth-order system of O D E s can be represented as a set of first-order quahtatwe dllterentlal constraints A

I

LmeaHv

independent

solutions

A standard result from the study ot ordmary ddferenhal equations is that lor a given linear system ot equations, the sum ot any two solutions to the system is also a solution This follows lrom the existence of linearly independent solutions We can state this precisely in the following theorem 1131 Theorem A.I. L e t the/tdnCtlOttS a~ a I , with a o ne~ep zepo (lo ~'

pt

+ (l l ~

I/ the set u,,, u s

L e t u,, u~, t]

]

+

, a,, he ( o n t m u o u s

on an interval 1,

u,, be tt ~ohltlOtl~ o f the n t h - o r d e t e q u a t i o n

+ tl ,,

l ~'

t

+ tl,, V

0

u,, ts hnearh, I n d e p e n d e n t , then every ~olutlon on 1 t~ o~ the

]olm ~ = C l l l I @ ( 2 l t 2 4.

w h e r e c ~,

c,, are constants

4

(.it

n

T h u s , flllV h n e a r c o m b i n a t i o n o/ 5olutzon~ z5 alao

a solution to the ttth-ordet equatzon

Quahtanve superposmon

195

A 2 QSIM systems are first-order In the general case we have an nth-order differential equation Such an equation can be reduced to a series of first-order equations, which each can be mapped onto a QSIM constraint This is stated in the following theorem taken from [10]

Theorem A.2. Let F[u(t), u'(t),

, u"(t)] = 0

be an ordmary dtfferentml equanon of order n, to be sansfied by a functton u [a, b]--~ R*, where F is defined only m terms of ad&non, mulnphcanon, and neganon, along wUh funcnons of contmuous and smctly nonzero denvanve Then a set of parameters and constramts can be defined, correspondmg wtth the equation, such that any reasonable funcnon u R---~g~ which sansfies the equanon also sansfies the set of constraints

Acknowledgement Many thanks to Ian Gibson and Dave Reynolds who helped clarify some of the quantitative mathematics Thanks also to the many kind people who, over a number of years, have contributed through discussions to the ideas presented here Ivan Bratko, Paul Brebner, Jason Chan, Paul Compton, Bart Feldman, David Hume, Bill Knelp, Ben Kuipers, and Claude Sammut I would also like to thank the two anonymous referees whose thoughtful suggestions have made considerable Improvements to the precision of the work presented here Early portions of this work were carried out while the author was a doctoral candidate at the Department of Computer Science, University of New South Wales, Austraha It was supported by a New South Wales Medical Research Training Scholarship, and was also assisted In part by a research award from the Medical Engineering Research Orgamsatlon

References [1] E W Coiera, Reasoning with quahtatlve disease histories for diagnostic patient momtorlng, Ph D Thesis, Department Computer Science, University of New South Wales, Kensington, NSW (1989) [2] E W Coiera, Momtorlng diseases with empmcal and model generated histories, Arttf Intell Med 2 (1990) 135-147 [31 R Davis, Diagnostic reasoning based on structure and behavlour, Arttf lntell 24 (1984) 347-410 [4] D Dvorak and B J Kulpers, Model-based momtoring of dynamic systems, in Proceedings HCAI-89, Detroit, MI (1989)

196

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V~ ( o/~/a

['~1 I dc Klcl.r and I ~, Brown A quahtatlx¢, ph~'qc~ b,e, cd on ¢_onflueni_e,, h u l lntd: 24 11984~ 7-8~ [6] k D Forbu', Ouahtatl'~c procc,,,~ theor', hit/ h m l l 24 (19~4) h~-16~ [7] k D Forbus The logic ot occurr~.nc.L. 1[1 Proceedings 11( 41-8" Nhlan IIah ( 1'4~71 -ll)~ 41"~ [8] k D Forbw, lmelhgent ~.omputei-md~.d engmt.ermg tl ~,tat~ 9 (19881 2"~ ~6 [q] P l Ha'~c ~, The NalxC Ph'~slcs Mamlc',[o m D MRhlc cd £ff~(tt S ~ t ( m , m the ~ltcloeletllotttt ~ge {Edinburgh Unlxer,,it~ Ih~, ,, Edinburgh %colldnd 19viI) [1111 B J Kulpcrs Quahtatlxc slmulatkm ttttt lntell 29 (i986) 2Nt~ ~,8 1111 B 1 Kulpels m d ( (hm f , u m n g intractabl~ I~ram.hlng in quahtalJ~c ~,ilntlldtlOll II] f'to¢~cdm~s IJ( 41-h7 Milan lhlh (It.~7) 1121 B J kmpt.rs %bqracllon by Iml~_-',cal~_ m qualltatl'~e ~,lmuI,Llkm m l'to~(~dlllk, I ~ I1 ,sS~.atlle ~ A 11987) 1~,] A L. R a b u n s l e m lnttc~ducl~ott to ()tdt/tat~ l)lffer(t~ttal Lqualton~ (,\~_adenu¢_ Prk~', No',~. 3 o r k 1072) 14] ~, M WcLss ( A k u h k o ~ s k ~ ,rod ~, Am,Hel A modal-based method lot tomt~ulc_r-,uded medical deep, ion making 4rtt/ h m l l 11 (1978) 14~ 172 IS] D S Weld ( o m p a r a t l ' , c ,mah,,qs 4tit~ lntell 36 (198;'~) z;?~, a7a