1 Introduction

DECOMPOSITION INTO INDEPENDENT DIAGNOSIS SUBPROBLEMS

Intelligent Tutoring Systems LaboratoryGlushkov Institutefor Cybernetics

40, Prospect Akademica Glushkova, 252022, Kiev,Phone: (044)-2676-934

E-mail : tsybenko@itsfreenet.kiev.ua

Abstract A decomposition of a complex probleminto independent subproblems is a standard methodto tackle large AI problems . This paper studies anotion of diagnosis independence related to model-based diagnosis and proposes a concept of P-decomposition of a diagnosis problem intoindependent diagnosis subproblems (IDS). We showthat a system P-decomposition depends critically onthe system structure and the measurementsperformed in the system. Taking into account boththese factors we introduce a notion of a conflictgraph and show that a system decomposition intoIDS may be reduced to the system conflict graphdecomposition.

This paper develops a formal algorithm forlocalization of multiple faults based on P-decomposition . We show that a strategy ofmeasurement selection for P-decomposition may beefficiently guided by pure "the first principleinformation" such as current observation and thesystem topology. In many cases the algorithm runsin logarithmical time complexity compared togeneral diagnosis time.

Model-based diagnosis (MBD) is a very dynamicand a wide field of research based on standard AItechnique such as predicate logic, heuristic searchand qualitative simulation. Current research in thisarea concerns a wide class of systems includingphysical and medical systems, student models, etc .Several practical applications, such as electroniccircuits, energy transport networks etc., have shownthe industrial potential of the approach .

In the recent years many new important results havebeen developed in the field of MBD. Among theseare: multiple behavioral modes [de Kleer andWilliams, 1989], focusing on probable diagnoses[de Kleer, 1991], physical negation [Struss andDressler, 1989] and physical impossibility[Friedrich et al ., 1990], focusing on independentdiagnosis problems [Freitag and Friedrich, 1992]and hierarchy design [Hamscher, 1990] .

Yury Tsybenko

Ukraine

However, the difficulties for building largediagnostic systems remains . One of the mainobstacles for building large diagnostic systems is thegreat number of possible diagnoses needed to beconsidered . Many real world applications deal withthe system composed of thousands of components .As examples let us consider energy transportnetwork and telecommunication network. As thesesystems are very large, diagnosis in such systems isa complex time consuming task and may cause highbreakdown costs . Practically such systems cannotwait until whole system will be diagnosed .

A standard approach to tackle a large AI problem isto decompose it into subproblems. Actually, thecardinality of the diagnosis set grows exponentiallyin the number of system components (systemdimension) . Reducing the system dimension weexponentially reduce the search space for diagnosis .Let us assume that suspected components arelocated within a certain subsystem . Is it possible toperform diagnosis only for suspected subsysteminstead of performing the whole system diagnosis?The lack of logically sound methods, excepthierarchy design, which allow to reduce diagnosisof the overall system to the diagnosis of itssubsystems is a major constraint on a widerapplicability of such an approach .

This paper studies a concept of diagnosisindependence and proposes a method ofdecomposition of a diagnosis problem into theindependent diagnosis subproblems (IDS). We showthat there is a certain set of measurements that,when performed, reduces the diagnosis of theoverall system to the diagnosis of its subsystems .We prove that the total diagnosis set is decomposedinto the Cartesian product of correspondent localdiagnosis sets. Proposed method does not rely on ahierarchy structure of a system to be diagnosed . Incontrast to hierarchy decomposition [Genesereth,1984], [Hamscher, 1990] this method is calledP(product)-decomposition.

In this paper we study the measurements that enableP-decomposition . Traditional GDE [de Kleer,

Williams 1987] approach applies probabilisticanalysis to decide what measurement to take next.This paper proposes a strategy of measurementselection which is efficiently guided by the systemdescription and current observation . To develop thisstrategy we introduce a notion of a conflict graph. Ithas been shown that P-decomposition of a systemmay be reduced to the system conflict graphdecomposition. A new interesting strategy for nextmeasurement selection based on a conflict graphdecomposition has been developed .

2 Motivation

This section presents two examples illustrating thecrucial points of our research. As the first examplewe use n=2k cascaded inverters.

Example 1. Suppose that the device input is 1 andthe output is 0 violating the prediction and tellingus that at least one of the components is faulted. Soadditional measurements are necessary to locate thefaults . If all gates fail with equal likelihood ahuman diagnostics should choose the inverter Ikoutput as the best next measurement . Whymeasuring Ik output is chosen as the best diagnosticaction instead of measuring another device point?What technique is the base for such an intuitions?

Figure 1 . n=2k cascaded inverters .

Measuring the component Ij, (j=l,n-1) output, nomatter what the value has been obtained as theresult of this measurement, divides the given systeminto two subsystems . Actually, component Ij outputis the output of the subsystem composed of thecomponents 11, . . .,Ij, and is the input of thesubsystem composed of the components Ij+1, . . .,In.As all inputs and outputs of both subsystems areknown the diagnosis of each subsystem isindependent of the overall system diagnosis. In ourexample measuring Ik provides the best (into equalparts) decomposition of the given diagnosis probleminto independent diagnosis subproblems .

At the first example we use the measurements todecompose a system into independent diagnosissubproblems . The second example illustratesanother approach to determining the IDS . It showsthat there is a situation when the IDS may be foundwithout probing, solely on the base of current

174

observation and the knowledge of componentbehavior .

Example 2 . Consider the binary circuit depicted onFigure 2 . If the input in](AND) is 0 then thecomponent AND behavior is independent of itssecond input in2(AND) . Moreover, in this case thediagnosis of the subsystem composed of thecomponentAND and the Device 2 is independent ofthe Device 1 diagnosis.

in1=0ll

ANDI--~'in2 =(l

FFigure 2 . AND is and gate, and Device], Device2

are arbitrary circuits

What is nature of the IDS for arbitrary system? Arethere common criterion for the IDS in aboveexamples? Are there efficient strategy for buildingthe II)S and how it may be applied to tacklecomplex diagnosis problems? The answer to thesequestions is the topic of the paper.

3 Preliminaries

This section follows the formal framework formodel-based diagnosis [Reiter, 1987] and [de Kleeret al ., 1992] .

Definition 1 . A system is a triple (Sd,Comps Obs)where : Comps, the system components, is a finiteset of constants ; Sd, the system description, is a setof first order sentences, defining the systemcomponent behavior and their connection ; Obs, aset of observations, is a set of first order sentences .

We adopt the Reiter's convention that AB(c) is aliteral which holds when component c e Comps isabnormal . We use the modeling stance that acomponent is abnormal if something is wrongphysically with it that may cause the incorrectoutput emergency. A model-based diagnosis for asystem is a conjecture that states: a certain set ofsystem components are abnormal, the othercomponents are normal . This conjecture mustsatisfy the system description and explain thesymptoms .

Definition 2 . Given two sets of components Cp andCn define D(Cp,C,d to be a conjunction :

[nAB(c) : c E CpJ n [n --AB(c) : c E Cn] .

A diagnosis for (Sd, Comps, Obs) is a formula D(A

,Comps-0) such that : Sdu Obs u D(A, Comps-A) is

satisfiable . A diagnosis D(0, Comps-0) is aminimal diagnosis iff for no proper subset Ol of Ais D(Al, Comps-Al) a diagnosis . We shall write Ainstead of D(A,

Comps-0), since subsetdetermines the diagnosis.

Definition 3. An AB-literal is AB(c) or -,AB(c) forsome c e Comps. An AB-clause is a clauseconsisting of AB-literals . A conflict of (Sd, Comps,Obs) is anAB-clause entailed by Sd u Obs . ConflictC is minimal if no proper subclause of C is aconflict for (Sd, Comps, Obs).

4 Diagnosis independence

So far an intuitive concept of diagnosisindependence has been considered . In this sectionwe formalizes such an intuition . Let us consider aformal statement that the diagnosis of a subsystemis independent of the rest of a system. In this casethe diagnosis of 'the rest of a system' is alsoindependent of the subsystem, in other words theconcept of diagnosis independence is symmetric.So, we have to consider a decomposition intoindependent diagnosis subproblems . What do wemean saying that "a diagnosis problem isdecomposed into independent subproblems A andB"? It means that for each local diagnoses for thesubsystem A its conjunction with a local diagnosisfor the subsystem B is a diagnosis for the system,and vice versa for each system diagnosis itsprojections on the subsystems A or B are localdiagnoses for the subsystems

4.1 Subsystems and local diagnoses

To introduce the formal definition of diagnosisindependence let as consider the notions of asubsystem and local diagnosis .

Definition 4. Let (Sd, Comps, Obs) be a system . Asubsystem is a triple (Sdl, Comps], Obsl), whereComps] is a subset of Comps. The subsystemdescription Sdl contains the axioms related to thecomponents from Comps], and the observationObsl contains the observation results related to thecomponents from Comps] . We shall writeSub(Cl__Ck) to indicate a subsystem composed ofcomponents C1, ._Ck . Each diagnosis for a (Sdl,Comps], Obsl) is called a local diagnosis forsubsystem .

Lemma 1 . Let (Sd, Comps, Obs) is a system, and Ais a diagnosis . A projection on a subsystem is alocal diagnosis for subsystem.

Proof. Let A1 be A projection on a subsystem (Sdl,Comps], Obs1) . If A is a diagnosis, then Sd u Obsu A is satisfiable . Hence Sdl u Obsl u A1 is alsosatisfiable, where Sd] c Sd, Obsl c Obs, O1 is asubconjunct of A containing AB-literals from A

related to the components mentioned in Compsl .

However, the contrary to lemma 1 proves to bewrong . The following example shows that there arelocal diagnosis such that their combination is not adiagnosis for the system .

Example 3. The familiar circuit (Figure 1) consistsof three multipliers MI,M2,M3 and two addersA1,A2.

Figure 3 . Familiar Circuit

The system description is given by:

ADDER(x) =:> {-AB(x) => Out(x)Inl(x)+In2(x))MULTIPLIER(x) =:>{ --. AB(x) =>Out(x)=1n1(x) *1n2(x) ) ;

Out(MI)=1n1(A1), Out(M2)=1n2(AI),Out(M2)=1n1(A2), Out(M3)=1n2(A2)-

The subsystem Subs(M2,M3) description containsmultiplier constraint, topology axioms : Out(M2)=Y,Out(M3)=Z, and a set of observation : In2(M2)=In](M3)=2, Inl (M2)=1n2(M3)=3 . Because thevalues of ports Y, Z are not known [0] is adiagnosis for Subs(M2,M3); [0] is also a diagnosisfor the subsystem Sub(Al,A2,Ml). However,conjunction [-,AB(M2) n -,AB(M3)] n [-%4B(MI) n--,4B(Al) n

4B(A2) ] is not a diagnosis for theoverall system .

4.2 Definition and criterion for diagnosisindependence

Definition 5. A diagnosis problem (Sd, Comps,Obs) is decomposed into independent diagnosissubproblems (P-decomposed) iff total diagnosis setfor (Sd, Comps, Obs) is the Cartesian product ofcorrespondent subproblem diagnosis sets .

The notion of a conflict is a basic notion for model-based diagnosis. The conflicts represent thediscrepancies between the predicted behavior andthe observed behavior, and form the basis for thederivation of the diagnoses . The following theoremgives a criterion of P-decomposition in terms ofconflicts .

Theorem 1. (Criterion of P-decomposition) Adiagnosis problem (Sd, Comps, Obs) is decomposedinto independent diagnosis subproblemsSub(Compsl) and Sub(Comps-Compsl) iff eachconflict for (Sd, Comps, Obs) is located within oneof the following subsystems Sub(Compsl),Sub(Comps-Compsl) .

Proof. (=:>) Suppose Sub(Compsl) and Sub(Comps-Compsl) are independent diagnosis problems. Theproof is by contradiction. Let us assume that there isthe minimal conflict C=LII v. . v Llk v L21 v. . vL2j, where L11--LIk are AB-literals forcomponents from Compsl, and L21, . ..,L2j are AB-literals for components from Comps-Comps] . Inthis case there is a local diagnosis O1 forSub(Comps]) such that AI contains LI]A . . .A Llk assubconjunct . Assuming to the contrary we obtainthat the clause LI1v. . vLlk is a conflict which is aproper subclause of the minimal conflict C. In thesimilar way we obtain that there is the localdiagnosis A2 for Sub(Comps-Compsl) containingL21n. . .A L2j as subconjunct. However, theconjunction Al n 02, contains all AB-literals fromthe conflict C , and hence, is not a diagnosis for(Sd, Comps, Obs)

(G) In accordance with [de Kleer et al ., 1992] theconjunct D(A,Comps-0) is a diagnosis for(Sd,Comps,Obs) if and only if it contains a kerneldiagnosis as subconjunct . Hence, it is sufficient toprove the theorem for the set of kernel diagnoses . Asimple algorithm of computing kernel diagnosis setfrom minimal conflict set has been proposed in [deKleer et al ., 1992] . Following this algorithm weobtain that a kernel diagnosis set for(Sd, Comps,Obs) is the Cartesian product of sets ofkernel diagnoses for subsystems Sub(Compsl),Subs(Comps-Compsl) .

Example 4 (Example 3 continued) . Additionalmeasurements are necessary for a decompositioninto independent diagnosis subproblems. Measuringpoint Y provides P-decomposition of the givenproblem into three independent diagnosissubproblems Sub(MI,A1), Sub(M2) andSub(M3,A2) . Actually, when the value of point Yhas been measured the correspondent I/O values forall above subsystems are known. So, the diagnosisof each subsystem maybe performed independently.

5 Strategy of P-decomposition

There are two polar strategies for determining theIDS . As it has been shown the IDS may be eitherinferred on the base of the knowledge of componentbehavior and current observations (Example 2) ordetermined by performing additional measurements(Example 1) . Does a general strategy for adecomposition into independent diagnosissubproblems exists? This section gives positiveanswers to these questions .

5.1 Conflict graph

In accordance with theorem 1 set of conflictslocated within correspondent subsystems isnecessary to decompose the given diagnosisproblem into independent diagnosis subproblems .Let us assume that a component behavior isdescribed by a set of constraints. Sets of constraintsrepresenting the different components are connectedby shared variables. The actual observation ismodeled as the value assignment to related variable .

Conflicts located within a certain subsystem arenecessary for P-decomposition. A conflict geometrydepends critically on the following parameters : 1) asystem topology ; 2) current observations . Todevelop a method of generating required conflictswhich takes into account 1) and 2) we applygraphical representation of a system structure .Consider a graph with two types of nodes: the set ofmain nodes Comps represents the set of systemcomponents, the set of auxiliary nodes Varrepresents the set ofshared variables, and the set Sdof graph edges represents component-variableconnections. We shall call the graph G a structuralgraph for the given system. A conflict graph for (Sd,Comps, Obs) is a minimal subgraph G(Obs) of Gcontaining those paths from G along which minimalconflicts for (Sd, Comps, Obs) have been derived.We assume that a conflict path begins at a mainnode and ends at a main node . Because the minimalconflict set depend on observations performed in asystem, the conflict graph G(Obs) represents thecurrent state of system observations .

5.2 Graphical criterion of P-decomposition

Let Compsl is a subset of components, defineSuspect(Compsl) is a subset of Comps] consistingof those components which are mentioned inminimal conflicts for (Sd, Comps, Obs) . Thefollowing theorem is a graphical analog oftheoreml .

Theorem 2. (Graphical criterion of P-decomposition) . A diagnosis problem (Sd, Comps,Obs) is decomposed into independent diagnosissubproblems Sub(Compsl) and Sub(Comps-Comps]) iff the sets of nodes Suspect(Compsl) andSuspect(Comps-Comps]) belong to differentcomponents (maximal connected subgraphs) of theconflict graph G(Obs) .

Proof. The proof follows from theorem 1 .

5.3 Conflict graph and observations

As we have seen in example 3 additionalmeasurements are necessary to perform P-decomposition . The measurements affect current setof minimal conflicts of a system being diagnosed.Consider a malfunctioning system composed ofmany components. The initial observation usuallyprovides a few minimal conflicts covering almost allof the system components . Minimal conflictsprovided by new measurements as a rule are propersubclauses of the early conflicts . Additionalmeasurements progressively reduce the size ofminimal conflicts, until faulty components havebeen located.

This section investigates how the measurementsperformed in a system affect the structure of thesystem conflict graph .

Example 5. Consider a circuit from example l . Theinitial observation provides single minimal conflictincluding all system components Cl, .-Ch .Correspondent conflict graph depicted on Figure 4is a line graph composed of 2n nodes . Measuringthe component Ck output provides new minimalconflict which is located within eitherSub(C], . . .,Ck) or Sub(Ck+I, . . .,Cn) . So, the resultingconflict graph may be reduced considerably .

In general, we cannot predict the structure ofresulting conflict graph until a measurementoutcome is not known . However, there is a conflictgraph invariant which does not depend on ameasurement outcome . The following theorems

characterize this important property of a conflictgraph .

Figure 4 . Conflict graph for cascaded inverters .

Theorem 3 . Let G(Obs) is a conflict graph for (Sd,Comps, Obs), and the auxiliary node x representsthe variable x. Suppose that the value of the variablex is known. Then the conflict graph G(Obs) doesnot contain any edge connected to the node x .

Proof. The proof is by contradiction . Assuming tothe contrary we obtain the minimal conflict C for(Sd, Comps, Obs) and at least the two components ci,cj mentioned in C which share the variable x. Aconflict may be derived in the following way :starting from the known values the inferenceprocedure propagates the known values through thesystem component constraints until theinconsistency has been detected . A minimal conflictis a minimal supporting environment for theinconsistency detected in the system [de Kleer,Williams 87) .

To prove the theorem we have to build the minimalconflict C1 such that Cl subsumes C andcorrespondent path for the conflict C1 derivationdoes not contain the variable x . To identify therequired conflict we begin our search at the emptyenvironment El, adding to it the components in thesame way that the conflict C was derived until thevariable x has been met . Because the value of thevariable x has been measured we can check theconsistency of El. If the observed value for x differsfrom the predicted value then the inconsistentenvironment El represents the required conflict .Otherwise, we can refine the environment El fromthose assumptions (components) that support thevalue assigned to the variable x. In this case at leastone of the components ci, cj must be deleted fromEl, contradicting to the fact that the conflict graphcontains a path between ci, cj.

Theorem 4 . Let G(Obs) is a conflict graph for (Sd,Comps, Obs), and the auxiliary node x representsthe variable x . Suppose that the component cbehavior does not depend on the value assigned tothe variable x . Then the conflict graph G(Obs) doesnot contain the edge joining the node c with theauxiliary node x.

The proof of theorem 4 is similar to the proof oftheorem 3 .

5.4 Selecting measurements for P-decomposition

To develop the strategy of measurement selectionfor P-decomposition we incorporate themeasurements in a 'conflict graph formalism' :measuring the variable x within a system entailsremoving all edges joining the variable x with theother conflict graph nodes from structural graph,and vice versa removing the edges connected to thevariable x points out that the variable x must bemeasured.

Suppose we have to decompose the given problem(Sd, Comps, Obs) into two IDS Sub(Compsl) andSub(Comps-Compsl) . To determine themeasurements that enable P-decomposition we usethe following Algorithm 1 .

Algorithm 1 .

Define G is a system structural graph, BorderVar isa set of border variables for Sub(Compsl) andSub(Comps-Compsl) .

1 . Loop until BorderVar is not empty :

2 . For each x e BorderVar DO:if the component c behavior does not depend on thevalue assigned to the variable x remove the edgesjoining the variable x with this component; generatenew BorderVar .

3 . IfBorderVar is not empty DO:select one variable x E BorderVar and measure x;delete all edges connected to the variable x; generatenew BorderVar; go to 2 .END.

5.5 Algorithm of P-decomposition forlocalization of multiple faults

The results obtained in the previous section providea formal background for a method for faultycomponent localization . The method is sequentialdecomposition of suspected parts of a system intoIDS and looks as follows :

Algorithm 2 (Informal algorithm ofsequential P-decomposition) .

Define SysList is a list of subsystems .

1 . SysList =(Sd, Comps, Obs) .2 . Loop until SysList is not empty .

3 . For every subsystem Sfrom SysList DO:

4 . Applying Algorithm 1 decompose the subsystemS into MS. Ifany discriminatory information is notavailable the best P-decomposition is provided by ahalf split.

5 . For every new subsystem SI DO:if the subsystem S1 consists of only one nodecorrespondent faulty component is isolated,otherwise add this subsystem to the SysList.END

Remark 1. If for each system component thevariable connected to its input may be measuredAlgorithm 2 gives single solution .

Remark 2. Each step of the algorithm prunesexponentially the diagnosis set . However, actualconflicts as well as actual diagnoses [Tsybenko, 94]are not lost .

Example 6 . Consider the conflict graph for thesystem depicted on Figure 4 . Because removing avariable node from this graph leads to the graphdecomposition, each measurement provides P-decomposition of the given system . The bestmeasurement is xk, because this measurementdivides the given system into equal parts . Let usassume that measuring the component Ck outputprovides new minimal conflict which is locatedwithin the subsystem Sub(CI, . . .,Ck) . In this case thesubsystem Sub(Ck+I_.,Cn) does not display themisbehavior so, it may be ignored . To locate theactual fault we have to repeat P-decomposition intoequal parts for the resulting conflict graph until oneelement conflict has been located . A half splitstrategy requires logarithmic time complexity ofgeneral diagnosis time for single fault localization .

5.6 Single measurement P-decomposition

Practical diagnosis requires the discovery of faultycomponents in a minimum number ofmeasurements. An important problem is to describethe case when single measurement provides P-decomposition . The following theorems 5, 6 provenin [Tsybenko, in prep.] present a solution of thisproblem .

If any discriminatory information is not available ahalf split strategy (illustrated in example 1) may beused in algorithm 1 for the next measurement

selection . Theorem 5 generalizes the techniquedemonstrated in Example 1 .Theorem 5. Let (Sd Comps Obs) is a system, and Gis the system structural graph . Suppose that thegraph G is acyclic . Then each single measurementprovides P-decomposition of the given system. Halfsplit strategy of measurement selection requireslogarithmical time complexity for localization ofmultiple faults.

Even if the structure of a conflict graph is verycomplex the single measurement P-decompositionexists. However, in this case algorithm 1 may run innon-logarithmic time . The following result holds inthe case of arbitrary conflict graph.

Theorem 6. Let (Sd Comps, Obs) is a system .Suppose that the variables connected to eachcomponent output can be measured . Then there is asequential P-decomposition locating the set offaultycomponents such that each step of the P-decomposition requires only one measurement.

5.7 Incorporating a preference criterionin the algorithm of P-decomposition .

Diagnosis of complex devices composed of manycomponents is difficult in part because the numberof possible diagnoses grows exponentially in thenumber of system components. It is unacceptable toconsider the set of all diagnoses . So, many diagnosisapproaches apply a preference criterion to restrictthe set of diagnoses needed to be considered . Thepreference criterion may be the number of suspectsin a diagnosis [Freitag and Friedrich, 1992] or theprobability of a diagnosis [de Kleer, 1991] . Apreference criterion allows to focus the diagnosisprocess on certain parts of a system.

The effectiveness of Algorithm 2 improvesconsiderably if a preference criterion is taken intoaccount . In this case each step of the algorithmconsists oftwo stages . At the first stage a preferencecriterion is applied to identify the suspected parts ofa system . At the second stage P-decomposition isapplied to separate these parts from the overallsystem . If the other parts of the system do notdisplay the misbehavior, they may be ignored . Toillustrate combined approach let us consider thefollowing example taken from [de Kleer, 1991] .Probability of a diagnosis is used as a preferencecriterion .

Example 7 . Consider n-bit adder (bI, ._bn) depictedon Figure 5 . Suppose that all inputs are 0 andoutput of the n-th bit On is 1 . Suppose that all gatesfail with equal probability. Focused GDE locates

probable diagnoses within the subsystemSubs(bn, bn-1), there are only five probablediagnoses (called leading diagnoses) : [Sl(bn.X])],[S1(bn.X2)], [Sl(bn-]A])], [Sl(bn-101)], [S1(bn-IA2)], where [SI(X)] indicates the candidate inwhich component X output is in mode output-stuck-at-1 .

Figure 5 . n-bit adder. XlX2 denote exclusive-or gatesA1,A2 are and gates, and 01 is an or gate .

Removing the edge connected to the point Cn-2 leadto the system structural graph decomposition intodisconnected subgraphs . Hence, measuring pointCn-2 decomposes the system into two subsystemsSub(bl, ._bn-2) and Sub(bn-1,bn) . Suppose that Cn-2=out(bn-2.01) is 0. So, the subsystemSub(b1, . . .,bn-2) does not display the misbehavior.Hence, it may be excluded from the diagnosis.Consider the structural graph for Sub(bn-I,bn)depicted on Figure 6 . Because one of the and-gatebn-1.A2 inputs is 0 the component bn-1 .A2behavior does not depend on the value assigned tothe variable bn-1 .x1 . So. correspondent edge isremoved from structural the graph Sub (bn-1,bn) .

Figure 6 . Structural graph for Sub(bn-I,bn)

The best next measurement is Cn-1 because itdivides the set of suspected components into equalparts . Suppose out(bn-1.01) is 0 . In this case thecomponent bn-A2 behavior does not depend on thevalue assigned to its output connected to thevariable bn.X1 . So, resulting conflict graph isdecomposed into three disconnected subgraphsG1=(bn-1_A1, bn-LA2, bn-1.01), G2=(bn.Xl,bn.X2), G2=(bn.Al, bnA2, bn.01) . The subsystemsSub(bn-1-Al, bn-I.A2, bn-1.01) and Sub(bn.Al,bn.A2, bn.01) do not display the misbehavior. So,they are ruled out from the diagnosis . Only thesubsystem

Sub(bn.Xl , bn.X2)

needs

to

be

decomposed . After measuring the component bn.Xloutput the diagnosis process stops .

6 Related works

Many work has been done to deal with thecomplexity of model-based diagnosis . Focused GDEapproach [de Kleer, 1991] applies componentfailure probabilities and focuses on the probablediagnoses . Struss and Dressler [Struss and Dressler,1989] use "physical negation" to rule out physicallyimpossible diagnoses . Hierarchy approach[Genesereth, 1984], [Hamscher, 1990], [Mozetic,1991] applies a multilevel hierarchical design thatallows to restrict the number of components to beconsidered at each level .

An idea of diagnosis independence has beenproposed in [Freitag and Friedrich, 1992] . Theyfocus on independent diagnosis problems (EDP) andpropose an algorithm for generating IDP . However,their concept of IDP ignores the symmetry ofdiagnosis independence, and their method does notapply the measurements for the IDP generating . Aformal analisys shows that a system P-decomposition entails a system decomposition intoIDP in meaning [Freitag and Friedrich, 1992]

7 Conclusions

The aim of this paper has been to reduce thediagnosis of a large system to the independentdiagnosis of its subsystems . A formal definition forP-decomposition of a diagnosis problem intoindependent diagnosis subproblems has beenproposed. However, we have not focused indeveloping purely theoretical results - instead wehave demonstrated how these results may be appliedin practical diagnosis. We have studied themeasurements that enable P-decomposition . It hasbeen shown that a strategy of measurement selectionmay be efficiently guided by "the first principleinformation" such as the system topology andcurrent observations . Hence it may be an alternativeto the traditional GDE approach . A formalalgorithm for localization of multiple faults basedon P-decomposition has been developed . In manycases the algorithm runs in logarithmic timecompared to general diagnosis time .

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