Abstract Hydro-ecologicalsystems comprisecomplexinteraction amongphysical, chemical,andbiological processes. Compositionalmodelingi,. e. creatinga system'sbehaviormodel byaggregatingmodelsof its constituents,is crucial for makingthe modelingtask feasible. Howevert,he composed modeils often too fine-grainedfor a particular task, for instance, in containingtoo manyirrelevant intermediate variablesor obscuringthe basic interdependenciesF.or this reason, the modelmayhave to be transformedand sim- plified. Thepaperpresents a graph-orientedrepresentation for dynamicsystemsclosely related to existing process languages,anda set of syntactic operatorsthat transform sucha modewl hilepreservingcertain propertiesof the model. Theformalismis motivatedandillustrated byan example taken from our workon modelinghydro-ecological systems,but wealso demonstrateits utility for technical applications.

Introduction

In our work on modeling complexecological systems for decision-support systems, a numberof important challenges arises. In particular, our efforts to obtain prediction models for algal blooms Rio Guatba (Southern Brazil), have to address problemsof: Ї Compositional modeling, i. e. generating a behavior model of a complex system through aggregation of models of its elementary constituents taken from a library. Ї Modeling of dynamicsystems, i.e. adequately capturing the evolution of the system and its phenomena over time, which in our application comprise a variety of processes from the flux of the river to chemical reactions. Ї Qualitative modeling in order to makethe essential distinctions only, thus enabling the modelingof classes of situations andthe exploitation of partial information, since both knowledge about the relevant types of processes and information about their specific instances is inherently imprecise, and available measurementsare sparse w.r.t, time andspace.

Several existing qualitative reasoning systems, such as

QPE ([Forbus

84])

and

QPC/QSIM

([Crawford/Farquhar/Kuipers 90]) have been built

satisfy these needs, and we wereable to formulate models of relevant processes in our domainusing QPC.However, weencounteredseveral difficulties that weconsider to be instances of general problems involved in compositional qualitative modeling of dynamicsystems. Several of them are related to the granularity of the model, seen from different perspectives: Ї Compositional modeling effects the structural granularity and there is the potential of generating overly detailed models. This is because the constituent modelsin the library have to be stated in terms of local variables and parameters only. The resulting modelcan be inappropriate both fromits cognitive and a technical point of view. Manyvariables and parameters maybe irrelevant fromthe perspective of the entire systemor a particular task and conceal the elementary influence structure of the system. Besides, somereasoning tools exhibit exponential behavior in the numberof variables, so it is desirable to keep the modelsmall. Ї Models of dynamic systems affect the temporal granularity of the model. It can be can be too fine- grained, again for both humansand predictive engines, if it captures all aspects of the dynamics.For instance, rapid but minimalfluctuations of the concentrations of some substances that are basically held in chemical equilibrium complicate the long term prediction of the behaviorof the entire system. Ї Qualitative modeling concerns the granularity of behavioral distinctions. Qualitative models can be too weakto derive all possible conclusions. This does not only concern the domainsof variables and parameters, but also the qualitative description of functional dependenciesis limited to monotonicfunctions, as is the case for manyqualitative simulators, the analysis of their counteraction or comparisonmaylead to spurious results. Someresearch has been carried out to address these issues: Ї Structural aggregation, particularly hierarchical modeling responds to the first problem by eliminating internal variables and parameters. Ї Behavioral approximation through distinction of time scales (sometimescalled time-scale "abstraction") aims

Heller 83

at suppressing the irrelevant details of temporal evolution ([Iwasaki 92], [Kuipers 87]). Ї Hybrid modeling attempts to introduce more distinctions through (semi-)quantitative information. This mostly confined to possibly repeated refinement of quantity spaces and algebraic operations (for instance [Williams88]). Our general view on the task is as follows. Often the following steps are distinguished and potentially supported bydifferent systems(Figure 1): Ї creation of a modellibrary, i.e. representing primitive modelfragments for a particular domain, Ї model composition, i.e. aggregating appropriate model fragments from the library to establish the modelof a specific system(this is, for instance, the task of QPC), Ї prediction/simulation, i.e. generating a description of the behavior of the entire system(e.g. QSIM'jsob). Weproposeto explicitly introduce a step of Ї model transformation that takes an initial model generated by compositionand transforms it into a model appropriatefor a particular task (Figure 1).

I

library creation I

task/

l

library

model cЇompositio=n -

osbjecive

~model

I = m°~'el=~t' , modeltransformation

prediction/simulation

s t~ran..s.f.o. rmed model

Figure 1: Overview of model-based prediction

This reflects our oppinion that a representation of knowledgerelevant to modelingin science and engineeing should distinguish between basic models and knowledge howto modify anduse these models, and that both types of knowledge should be represented in a formal and declarative way.This is different formother approachesto compositional modeling which presume in some way that all elements needed for an adequate compositional model are already "pre-manufactured" and simply need to be collected from the library. Wedo not believe that it is feasible to producein advanceall kinds of combinationsof abstractions, approximations, and simplifications of the basic models. The presentation of our work focuses on the transformation step. This means, we assume an existing composedmodelof a system as an input to our formalism which then generates a newmodel under preservation of certain properties. There are two contributions in this work:

Ї a modelinglanguage for specifying influences and Ї a set of local syntatic operators that transform models expressed in this language. In order to improve both the predictive power of the modelsand the results of the operators, the confinementof qualitative functional relationships to monotonicdependencies only has to be overcome. A commonanswer to ambiguityand insufficient distinctions in quantitative models is "Hybrid modelsby integrating quantitative information!". Webelieve that this seeminglyobvious solution is often inappropriate and obscures the fact that the expressive power of the models can be extended without havingto leave the realmof qualitative descriptions. Our modeling language extends the expressive power of languages like QPE/QPCin allowing for Ї more distinctions between functional dependencies (than just monotonicand algebraic ones) and Ї moregeneral types of influence combinations (than just linear combination). Weillustrate the application of the language using an example from the hydro-ecological domain. Weaim at a formal characterization of model relations and transformations is required a) for an automatedsolution to the problem and b) for determining the impact of the transformation, i.e. the properties gained and the properties preserved by the transformation, whichis required. Theset of operators includes Ї generationof strict abstractions of a modelas well as s approximation of dynamic relationships through functional dependencies These transformations are independent of the quantity spaces chosenfor the variables and parameters (in contrast to [Williams]). Theuse of the operators is illustrated by the ecology exampleand, to demonstratethe versitility of our formalism, by an example taken from a technical domain. Wecontinue by giving a brief introduction to the hydro-ecology background. The Problem Domain In an international collaboration betweenresearchers of Brazil, France and Germany, we have been examining a specific ecosystem, namely the Rio Guafba in Southern Brazil, with the objective of analyzing and predicting undesirable occurrences of algal blooms. The modeling of the complexhydrodynamicsand the various chemical and biological processes involved provided us with important challenges for our modelingand reasoning techniques. Amongthe elementary conditions for the possibility of algal bloomsis the availability of nutrients, which is influenced primarily by distribution and transformation processes. In this paper, we will examine a typical exampleof an interaction of twosuch processes.

84 QR-96

Advection Advectionis the transport of matter by directed flow of water. The complexhydrodynamicsin the bays of the Rio Guafbaprevent us from using a linear water flow model and wehad to choose a flexible repesentation of spatial distributions and water transportation. By using compartments, the elements of a topological partitioning of the water body(described belowand in moredetail in [Heller 95]), and by locating the transport processes between adjacent compartments,we also gain moregenerality. The advective effect on the concentration of somespecific chemicalconstituent in twoadjacent compartmentscan be easily determined if the volumesare assumedconstant (requiring the net flow for each compartmento be zero). Asimple modelunder this assumptionis discussed below. AmmoniaDissociation One of the most important constituents is ammonia, appearing both in free (NH3) and ionized form (NH4). Both forms can act as nutrient, but free ammoniain high concentrations can also exhibit toxic effects, so wehave to study the chemical equilibrium (NH4+ OH~ NI-I3 + H20) established by the counteracting reactions of ionization and dissociation. Both reactions are strongly influenced by the pHof the location. Toput it moreprecisely, if the ratio between(the molar concentrations of) NH3and NI-h is below 10(PH-9.26), then the dissociation reaction dominatesionization. Abovethe given reaction constant, the ionization is predominant. Both reactions will be modeledas a single process with a rate that is linearly dependent on the difference between the ratio NH3/NHa4nd the reaction constant (modeledas positively monotonicin the pH, see Figure 7). Model Representation We present a modeling language with a flexible representation of functional and integrative influences and we depict models in this language by using a graphical notation, which will help to illustrate the examples throughout the paper. Asystemmodelconsists of a finite set of variables with Ї continuous real-valued functions over time (or any appropriate qualitative abstraction thereof) as domains. There is a set of constraints on these functions, represented by the existence of "influence functions" specifying the dependencyof a variable on a set of other variables. Characterization and Combination of Influences The basic influence function is a multivariate monotonic function with multiple parameters. In particular, wewant to express that a variable, A,dependsmonotonically on a

set of other variables, {Bt, B2..... Bn}(possibly with different direction coefficients, $1, $2..... Sne {+1, -1}). Formally: 3fe Mon(sl,s2,...sn)Vte A(t)= f(Bl(t), B2(t)..... wherefe Mon(st,s2,...sn)iff Vie{ 1,2 ..... n f(xl..... xi ..... x.) >f(xl..... xi', .... x.) ў:~(Si"xi >Si" with Sic {+1,-1} (l

Integrative Influences The discussed influences correspond to the qualitative proportionalities used in QPT([Forbus 84]). To represent the so-called "direct influences" of QPT, we need an integrative influence, expressing that the derivative of a variable A is (monotonically) dependent on a group other variablesBl, B2..... Bn: dA(t) d---~= f(B, (t) ..... B) e(t)), f ў Mon(sl,s2,...sn Weuse all of the constructions of function restrictions and decomposition discussed above. In graphical display we enclose the combination information in a circle or a rectangle with roundededges, as is shownin Figure 5. ~-qs ...

n

m

B = ~fi(Ai) -~gj(Cj)

i=l

j=l

fi eFt (l~i

Analogously we can decompose (single) influences multiplicatively. For more sophisticated constructions, intermediate variables have to be used. The graphical notation uses two newcombination symbols("x" and "/", see Fi :ure 4).

II 1-Ifi (Ai) B ~ i--I figj(ej) j=l

-,°ў1 ficFi (1-

dB= f(Ai,A2 .....

dt

An), fEMon(st...s2)

with the additional restrictions fromthe F~

Figure 5: Integrative influences

In this waywecan represent a qualitative abstraction (with respect to the functions involved) of an ordinary differential equation in our modeling language and in turn extract a partially specified differential equation froma diagram. Together with a mechanismfor instantiating and composingmodel fragments, we can also visualize models written in QPE/QPCnotation.

Process Models for the Domain Problem Processes are described by partial influence diagrams (possibly with parameters) and additional information about howto composethem with other processes acting on commovnariables. Aprocess is instantiated by giving the parameters defined values that can be obtained from the system description and aggregating the partial diagrams into the system model. The formal semantics described in the last section dependon the closedness of the model. In the cases discussed here (transport and chemical transformation), we have apparently additive combination of influences. Wedeveloped models for prediction tasks for both the short and the long term behavior of hydro-ecological systems. Here we will present 0nly two simple ones to studytheir interaction. As a basic modelingdecision, wedivided the water body of the river under consideration into compartments,i. e.

86 QR-96

regions with similar flow characteristics, that are assumed to have homogenoups arameter values. The partitioning is a spatial abstraction that preserves only topological information (basically the neighborhood relation) and some individual properties of the compartments(e. g. volume). Variables are associated with single compartments(e. g. ammoniaconcentration) or a set of compartments(e. the directed water flow betweenadjacent compartments). A simple generic process description for advection of someconstituent (e. g. ammonia)between two adjacent compartments (src and dest) is shown in the influence diagramin Figure 6.

[constituent [src]

rate.adv ~ constituent [const, src, dest]~ [dest] |

Figure 6: The advection process (simple version) Unlabeled arrows are to be read as bearing the identity label "Id '°. The boxes with a black shadow denote important state variables. Theyrepresent concentrations. Thus, the transported amount of matter is obtained by multiplication of the source compartmentconcentration with the (absolute) flow between the compartments. The loss respectively gain in concentrationis then calculated as a linear function (the linearity factor being in either case the reciprocal of the volume of the respective compartment,which is assumedconstant). It will instantiated for various chemical constituents, const(ituent), and locations, src and dest. Note that the semanticsin the strict sense givenin the last sections will be valid for the complete (composed) model onlyЇ

NH4[location]

reaction-constant. d~ssocl[ocation] Lin

Figure 7: Dissociation process without feedback However, the combination of the influences on the concentration in the destination compartmentare assumed to be additively decomposablefromother influences. Furthermore, a version of the dissociation process without feedback will be used. The concentration of NH4 will be treated as equaling the total ammoniaconcentration. Thus, we can neglect the loss of NH4by the transformation. The resulting influence diagram is presented in Figure 7. Model Composition Wecomposethe advective transport of ionized ammonia (whichwe treat as total ammonias, o that NI-I3 is assumed not to be subject to advection) fromcompartment"In" into a specific compartment, X, and from X to compartment "Out", with the dissociation taking place inside compartment X, weobtain the diagramin Figure 8. So we benefit from being able to composethe system modelfrom a simple structure description and a library of generic process descriptions (both described in detail in [Heller 95]), but the simulation of the resulting modelis unnecessarily complicated by the large difference in the

(In)

i compartmentX !

I

I, NH4[In] ~ ad~~e'in,X] P'~~LinNI~tX]

I~ (Out) I

[.,in

rate.

~- adv[NI-h,X,Out]

I.n°wtIn,xlI percentage. I ~ ra :e. [-xa~ reaction-constant.

[ dissoc[X][~n-~-[disscc[X] ~ dissoe[X]

I

Lin

M~

Ї Figure 8: Interaction of advection and ammoniadissociation

HeUer 87

strength of the integrative influences. Somequalitative simulation frameworkslack a wayto express the different orders of magnitude and therefore even produce spurious solutions by erroneous assessmentof the counteraction. For testing purposes, we transformed the obtained influence diagram into the modeling language of QSIM ([Kuipers 86]), like QPCwould do. Unfortunately, loose the causal information represented in our models, which is partially responsible for some problems of efficiency. A part of the QSIMalgorithm exhibits exponential behavior in the numberof involved variables. From the misjudgement of the relative orders of magnitude of the effects of transport and transformation also impossible behavior branches resulted. Even for slowly rising NH4values, the ratio of the concentrations is hypothesizedto be significantly out of equilibrium. For the illustration of this effect, an extendedexampleis given in [Heller 95]. Wepropose a solution that will will both reduce the number of variables and make use of the information aboutthe different orders of magnitudein the effects of the interacting processes to rule out spurious solutions. This will be achieved by local syntactical operators transforming a given influence diagram. Model Transformations Wedevelopeda set of transformation operators to simplify influence diagramsin order to identify the basic influence structure in more complexinteractions. The goal is to examinein a formal waythe applicability of the so-called time-scale abstraction. Time-scale abstraction, as introduced by BenjaminKuipers ([Kuipers 87]), will formally be treated as an approximation. In the formal framework of model relations developed in [Struss 92], abstraction transforms a modelinto a strictly weakerversion, whereas approximation replaces one model by another one that mayviolate validity. Abstraction Operators Anoverviewof abstraction operators is shownin Figure 9 (on the next page). If a variable specified in a model fragment is assumed constant in the context of the complete model, we can eliminate it, because it unnecessarily complicates the reasoning task. The elimination of constants is achieved by the operator (9a). The class of functions on the right hand side is obtained by taking the maximumwith respect to set inclusion (rememberthat Id c Linc Lip c Men). The resulting modeltransformation is an abstraction (even moreprecisely, a "view" as defined in [Struss 92]). The proof for this operator and for the following ones can be foundin [Heller 95]).

Somevariables might be irrelevant, e. g. because they are not observable. The elimination of intermediate variables for multiple influences is shownas (9b). Analogousoperators exist for integrative influences, on somevariable Ci. For an integrative influence onB there is a restriction (at least in the semacticsused): Bcan only eliminated, if all of the influence originating from B are linear (see 9c). Partial decompositionof influences can be preserved, if the relating function is linear. For completely decomposableinfluences, the operator has the form shown in (9d). Various cases with additional influences on C, not originating from the intermediate variable B, are consideredin [Heller 95], but will not be discussedhere. All of the operators above reduce the number of variables, whichis an advantagein itself. Another class of operators achieves the subsumption of parallel influences, i. e. of influences with the same source and destination and the same combination symbol (either "+" or "-"). Figure 9e showsthe decomposedcase, whichis the simplest one. Time-Scale Abstraction as Approximation To cope with widely separated time-scales and to makethe reasoning task feasible in cases where "fast" and "slow" processes interact, weintend to identify subsystems (by employingthe operators introduced above) that can - under certain conditions - be substituted by functional dependencies, while committing only a neglectable approximation error. This corresponds to the technique of "abstraction by time-scale" as defined in [Kuipers 87]. If the elementaryinfluence structure has one of the following forms, weuse the solution of the equilibrium equation as substitute. In Figure 10 we show two operators acting on closely related structures, namelyon direct linear self-stabilization (10a) and on multiplicatively mediated linear selfstabilization (10b). Both are discussed in detail [Heller 95]. In the first case it is even possible to derive precise bounds on the approximation error committed, by analysis of the underlying ordinary differential equation. In general, the quality of the approximation increases with the linearity factor of the stabilizing function (class FB) and decreases with the Lipschitz coefficient of the transfer function (class FA) and the maximumvariation of the derivative of A. So we profit from preserving the information about the function class restrictions (namely linearity and the Lipschitz condition) while using the abstraction operators shown in Figure 9. At this point the additional information represented pays off.

88 QR-96

9a) Eliminationof (multiplicative) constants: ~Lin 9b) Elimination of intermediate variables (with multiple influences):

FAЇ { Id, Lin, Lip, Mon} Se Ix,/}, K constant FA'= max(Lin, F,) S' = sign(K)e 1+,-} Fi,Fj' Ї { Lin,Lip,Mo}n, Si,Sj' Ї {+,-} (1 _

FAЇ {Id, Lin, Lip, Mon} FBЇ {Id, Lin} Sa, SBЇ {+, -1 F'= max(FA,FB) S' = S^. SB

Fi Ї {Lin,Lip, Mon(}1 _

FI, F2Ї {Id, Lin, Lip, Mon} Sl = S2Ї {+,-} F' = max(Fl, P2,Lin) S'=SI (=$2)

Figure 9: A selection of basic abstraction operators for influence diagrams

Heller 89

10a) Time-scaleabstraction for linear self-stabilization: FB 10b) Time-scaleabstraction for multiplicatively mediated linear self-stabilization (with an additional influence):

FAe {Id, Lin, Lip} Fa ~ {Id, Lin} FA'= max(FAF, B)

FAe { Id, Lin,Lip} Fpў{Id, Lin, Lip, Mon} FA'= max(FAL,in) FD'= max(FDL,in)

Figure10: Twotime-scaleabstractionoperators for influence diagrams

Application to the Example Model Touse this kind of approximationfor the modelgiven in Figure 8, we have to identify the elementary influence structure of the faster subsystem.Therefore, the variables reaction-constant.dissoc[A] and then rate.dissoc[A] are leiminated using the the operator from Figure 9b, which yields the influence diagram on the left hand side of Figure 11. The influence structure that appears nowin the lower part of the figure is a case of a multiplicatively mediated linear self-stabilization. It will be approximatedby using the operator shownin Figure 10b, whichis justified by the strong stabilization by the chemical reaction and the comparativelyslow changesin NH4(the effects differ by factor of about 10"). The backgroundknowledgeabout the orders of magnitudeof the influences can be attached to the model fragments by the modeler (and propagated consistently through all abstraction operations), so the decision about the application of the approximation

operator can be taken by formal reasoning about local information. So the simpler model on the right hand side of Figure 11 can be used for purposes of middle and long term prediction with a substantial increase in efficiency. In our test runs with QSIM, we obtained much more focused predictions (usually a single one instead of morethan 10 behavior branches) and all truly spurious solutions were ruled out. Another Example: Motor with Control Circuit Wegive a short example for the use of the modeling languageand the transformation operators in the technical domain. Wehave modeled a direct current motor with control circuit (described in more detail in [Malik/Struss96]). The influence diagrams of the components were derived directly from the following Differential Equations (for the parameterdescriptions refer to the table below):

rate adv I~ [NI'I411u,A]

Lin, _l_"l

~~ [N~e,)~.a~tut]

]

] flow[A,Out] ~

l: ....... IF-~

rate.adv Lin [NH4,1n,A]

l~'llJ.rA'l -- LLinin, ra1t^et.adv I

~ [Nl-h,A,Out]

[

~Lin

Figure11: Theexamplme odel.Left side: after the eliminationof intermediatveariables, Rightside: after applyingtime-scaleabstraction 90 QR-96

v = 2d- to m CC

tom= (1-S).co

dco = cM.v - co

dt

T

(controller) (sensor) (motor behavior)

In Figure14the result of further elimination(of tomand then v) is shown: Lin

v drivingvoltage

CO rotationalspeed(of the motoraxis)

Lin~

oh measuredrotational speed d desiredrotational speed

Figure 14: The motor model after the second

Cc controller constant

simplification

CM motorconstant T motorinertia S slip (of the measurinpgulse wheel) Froma simple structure description, the following influencediagramwill be derived(Figure12):

Finally, subsumptioonf the resulting parallel influences (operator9e) identifies the elementaryinfluencestructure as a (direct) linear self-stabilization that can approximatedby a functional dependency(see Figure 15, TSA-operatofrromFigure 10a):

Applicationof the developedoperators eliminates the

Lin

constants(inclusively the derivedconstant l-S) fromthe

model(operator fromFigure9a) and also successivelythe

intermediatevariables2d, 2d-corn,CMVdo, J/dt, andCMV--CO

(operators shownin Figure 9, b throughd) yielding the

simplermodelin Figure13.

.........

. ..........

Ї ..............

Lin ~ '

Figure 15: The f'mal time-scale abstraction of the motor model

Ї ............

° ........

°°°. ........

!

Figure 13: Motormodelafter the first simplification

In the desired case (no slip: S = 0, controller constant equals motorconstant: Cc= CM),the approximationerror can be bounded by V=.T.c, c being a bound on the derivative of d, thus showingthe response of the motor beingdependenstolely onthe inertia. Fordetails refer to [Heller95].

tor_

2d-

"

~

~

! Controller

........

-=1

.......................................

i, ........................

J

Figure12: Structuredescription and influence diagramof a motorwith control circuit

Heller 91

Future Work Whatwe have achieved at the present time, is an initial theory of transforming models of dynamic systems. The sets of operators developedand provedso far is certainly not complete. Wealso intend to introduce more function classes, for instance, in order to determine dominant influences in a combination of counteracting ones (e. g. overlinear versus linear growth). Anothertheoretical issue is to analyze fixed points of the application of the set of operators. Agoal wouldbe a guarantee for deriving some normal form of a model, at least w. r. t. informationabouta specific task andobjective of analysis. This raises the issue howto represent such information and how to control the application of the operators (as indicated in Figure 1). An implementation will be done probably based on a hypergraph grammarapproach. Such a model transformation moduleinteracts to somedegree with its neighboring tasks. Althoughthe transformedmodelcould, in principle, be fed to a system like QSIM,the increased power of our modeling language in terms of characterization of functional dependencies and the preservation of causal information will presumably also lead to stronger predictors. On the other hand, although a model stated in QPE/QPCcould be an input, this would lack the distinctions necessary to achieve the strongest possible results. Weconsider to the graphical notation introduced in this paper for interactive definition of modelfragments and composedmodels. Somefundamental issues about the specification of modelfragments have to be examinedfurther. So far, the semantics are defined only for the completely composed (and closed) model. However, compositional modeling requires a formulation of isolated fragments in the first place, and it is not obvious what such a "context-free" model fragment "knows" about the appropriate way of combiningwith other fragments. For some class of processes, like transportation and transformation processes, it is evident, that the effects combineadditively with other processes. Possibly, the ontologyhas to be extendedby a classification of processes which, based on their physical nature, uniquely determines the correct type of combination. Studies in other domains and with different exampleswill shed a light on different mechanismsfor combining processes influencing shared variables.

Acknowledgements Wewouldlike to thank for the valuable collaboration with Francois Guerrin and Waldir Roque. This work was partially supported by the Brazilian Research Council (CNPq) and the German Ministery of Education and Research (BMBF).Webenefitted from discussions with Paulo Marcos Amaral Alves, Elenara Correa Lersch and Maria Mercedes Bendati from the research department of DMAEF.inally we wouldlike to thank the reviewers for their helpful remarks.

References

[Crawford/Farquhar/Kuipers 90]

James Crawford,

AdamFarquhar, Benjamin Kuipers: QPC:A Compiler

from physical models into Qualitative Differential

Equations. AAAI1990.

[Forbus 84] Kenneth Forbus: Qualitative Process Theory. In: International Journal of Artificial Intelligence 24 (1-3), 1984.

[Heller 95] Ulrich Heller: Temporale Verhaltensabstraktion am Beispiel yon Regelkreisen und hydro- 6kologischen Systemen. Master Thesis, Department of Computer Science, Technical University of Munich, Germany, 1995. (in German)

[Heller/Struss 96] Ulrich Heller, Peter Struss: Qualitative Modeling for Environmental Decision Support. Symposium Informatik fiir den Umweltschutz, Hannover, Germany, 1996.

[Iwasaki 92] Yumi Iwasaki: Reasoning with Multiple Abstraction Models. In: Boi Faltings, Peter Struss (eds.): Recent Advancesin Qualitatives Physics, MIT press, 1992.

[Kuipers 86] Benjamin Kuipers: Qualitative Simulation. In: International Journal of Artificial Intelligence 29(3), 1986.

[Kuipers 87] Benjamin Kuipers: Abstraction by TimeScale in Qualitative Simulation. AAA1I 987.

[Malik/Struss96]

Andreas Malik, Peter Struss:

Diagnosis of DynamicSystems Does Not Necessarily

Require Simulation. QR1996.

[Struss 92] Peter Struss: What's in SD? - Towards a Theory of Modeling for Diagnosis. In: Hamscher, Console, de Kleer (eds.), Readings in Model-based Diagnosis, MorganKaufmann, 1992.

[Struss 93] Peter Struss: On Temporal Abstraction in Qualitative Physics - Apreliminary report, QR1993. [Williams 88] Brian Williams: MINIMA:A Symbolic Approachto Qualitative Reasoning, AAAI1988.

92 QR-96

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