Knowledge Compilation Techniques for Model-Based Diagnosis of Complex Active Systems

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Knowledge Compilation Techniques for Model-Based Diagnosis of Complex Active Systems

Gianfranco Lamperti, Marina Zanella, Xiangfu Zhao

To cite this version:

Gianfranco Lamperti, Marina Zanella, Xiangfu Zhao. Knowledge Compilation Techniques for Model- Based Diagnosis of Complex Active Systems. 2nd International Cross-Domain Conference for Ma- chine Learning and Knowledge Extraction (CD-MAKE), Aug 2018, Hamburg, Germany. pp.43-64,

�10.1007/978-3-319-99740-7_4�. �hal-02060059�

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Knowledge Compilation Techniques for Model-Based Diagnosis of Complex Active Systems

Gianfranco Lamperti¹, Marina Zanella¹, and Xiangfu Zhao²

1 Department of Information Engineering, University of Brescia, Italy

2 College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, China

Abstract According to complexity science, the essence of a complex system is the emergence of unpredictable behavior from interaction among components.

Loosely inspired by this idea, a diagnosis technique of a class of discrete-event systems, called complex active systems, is presented. A complex active system is a hierarchical graph, where each node is a network of communicating automata, called an active unit. Specific interaction patterns among automata within an active unit give rise to the occurrence of emergent events, which may affect the behavior of superior active units. This results in the stratification of the behavior of the complex active system, where each different stratum corresponds to a different abstraction level of the emergent behavior. As such, emergence is a peculiar property of a complex active system. To speed up the diagnosis task, model-based knowledge is compiled offline and exploited online by the diagnosis engine. The technique is sound and complete.

Keywords:Knowledge Compilation; Complex System; Emergent Behavior; Discrete- Event System; Active System; Model-Based Diagnosis; Lazy Techniques.

1 Introduction

In an interview in January 2000, the physicist Stephen Hawking was asked the following question [32]:Some say that while the 20th century was the century of physics, we are now entering the century of biology. What do you think of this?Professor Hawking replied:I think the next century will be the century of complexity.Colloquially, we use the qualifier “complex” to indicate something that is complicated in nature. In this perspective, acomplex systemis complicated in structure and behavior, such as an aircraft or a nuclear power plant. However, according to complexity science [1,2,24,8], although it is likely to be complicated, a complex system does not equate to a complicated system. In this different perspective, a complex system is composed of several individual components that, once aggregated, assume collective characteristics that are not mani- fested, and cannot be predicted from the properties of the individual components. So, a human being is much more than the union of some 100 trillion cells that make up her body. Likewise, a cell of a human body is much more than the union of its molecules.

What we think as a human being, in terms of personality and character, is in fact a collective manifestation of the different interactions among the neurons and synapses in the brain. On their turn, these are in continuous interaction with the other cells of the

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body, possibly with clusters of cells that constitute organs, which themselves are complex systems. Locally, each cell is characterized by its specific behavior and interaction rules, globally resulting in the collective manifestation of the human being.

Complexity science questions the traditional reductionist approach adopted in the natural sciences, namely the reduction of complex natural phenomena to several simple processes, and the application of a simple theory to each process. More specifically, the principle of superimpositionis no longer accepted, namely that the comprehension of the whole phenomenon relies on the superimposition of its parts. Based on this principle, for instance, if you understand the theory of an elementary particle, then you will understand every natural phenomenon. Likewise, if you understand DNA, then you will comprehend all biological phenomena. By contrast, a significant aspect of complex systems is that a new collective level emerges through interactions between autonomous elements. Hence, in order to comprehend the complex system, additional knowledge is required beyond the comprehension of the single elements. More generally, a complex system is structured in a hierarchy of semi-autonomous subsystems, with the behavior of a subsystem at leveliof the hierarchy being affected, although not completely determined, by the behavior of each subsystem at leveli−1. As such, the behavior of a complex system isstratified.

Loosely inspired by complexity science, this paper presents a method to extract knowledge - above all, about emergent behavior - from the models of individual clusters of (component) systems and to exploit this knowledge for the lazy diagnosis of a class of discrete event systems (DESs) [5], calledcomplex active systems (CASs). A sort of CASs was first introduced in [21,20]; however, the relevant model-based diagnosis task proved naive. Subsequently, a viable diagnosis technique was presented in [17,23]

and extended in [22]. To the best of our knowledge, apart from the works cited above, no approach to diagnosis of DESs (much less to diagnosis of static systems) based on the complexity paradigm has been proposed so far. Moreover, none of the complexity- inspired approaches cited above is based on knowledge compilation. Still, several works can be related to this paper in varying degree, as discussed below.

An approach is described in [14], where the notion of asupervision patternis introduced for flat DESs, which allows for a flexible specification of the diagnosis problem.

However, the events associated with supervision patterns are not exploited for defining any emergent behavior. Diagnosis of hierarchical finite state machines (HFSMs), which are inspired by state-charts [9], provides a sort of structural complexity. Diagnosis of HFSMs has been considered in [13,26]. However, complexity is merely confined to structure, without emergent events or behavior stratification. An algorithm for computing minimal diagnoses of tree-structured systems is presented in [31]. Subdiagnoses are generated by traversing top-down the tree, which are eventually combined to yield the candidate diagnoses of the whole system. However, despite the fact that the considered systems are static and the diagnosis method is consistency-based, neither complexity nor emergent behavior is conceived, as the goal of the technique is efficiency of the diagnosis task by exploitation of the structure of the system. An approach to consistency- based diagnosis of static systems supported by structural abstraction (which aggregates components to represent the system at different levels of structural detail) is described in [6] as a remedy of computational complexity of model-based diagnosis. Evidence

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from experimental evaluation indicates that, on average, the technique performs favor- ably with the algorithm of Mozetiˇc [25]. Still, no emergent behavior is conceived. A technique for consistency-based diagnosis of multiple faults in combinational circuits is presented in [30]. To scale the diagnosis to large circuits, a technique for hierarchical diagnosis is proposed. An implementation on top of the tool presented in [12], which assumes that the system model has been compiled into propositional formulas in de- composable negation normal form (DNNF), has demonstrated the effectiveness of the approach. However, neither emergent behavior nor behavior stratification is conceived, and the technique addresses static systems only. A scalable technique for diagnosability checking of DESs is proposed in [28]. In contrast with the method presented in [27], which suffers from exponential complexity, new algorithms with polynomial complexity were proposed in [15,33], called thetwin plantmethod. However, the construction of a global twin plant, which corresponds to the synchronization based on observable events of two identical instances of the automaton representing the whole DES behavior, is often impractical. The method proposed in [28] exploits the distribution of a DES to build a local twin plant for each component. Then, the DES components (and their relevant local twin plants) are organized into ajointree, a classical tool adopted in various fields of Artificial Intelligence, including probabilistic reasoning [29,11] and constraint processing [7]. The transformation of the DES into a jointree is an artifice for reducing the complexity of the diagnosability analysis task. Neither emergent behavior nor behavior stratification is assumed for the DES, nor any knowledge extraction is performed. A variant of the decentralized / distributed approach to diagnosis of DESs is introduced in [16], with the aim of computing local diagnoses which are globally con- sistent. To this end, as in [28] but in the different perspective of diagnosis computation rather than diagnosability, a technique based on jointrees (calledjunction treesin the paper) is proposed. The goal is to mitigate the complexity of model-based diagnosis of DESs associated with abduction-based elicitation of system trajectories. However, the goal of [16] is the efficiency of the diagnosis task, which is performed online only, thereby without exploiting any compiled knowledge. The transformation of the DES into a junction tree is a technical stratagem for improving the decentralized / distributed approach to diagnosis of DESs. The DES is plain, with no emergent behavior.

The contribution of the present paper is threefold:(a)specification of a CAS based on active units,(b)proposal of a process for extracting knowledge from active units, and(c)specification of a diagnosis task for CASs exploiting compiled knowledge.

2 Complex Active Systems

A CAS can be defined bottom-up, starting from its atomic elements, theactive components(ACs). The behavior of an AC, which is equipped with input / outputterminals, is defined by a communicating automaton [3]. The transition function moves the AC from one state to another when a specific input eventis readyat an input terminal. In so doing, the transition possibly generates a set ofoutput eventsat several output terminals. If specified so, an AC can perform a transition without the need for a ready event;

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formally, the transition is triggered by the “ε”emptyevent.³ ACs communicate with one another throughlinks. A link is a connection between an output terminalo of an ACcand an input terminali⁰of another ACc⁰. When an eventeis generated atoby a transition inc,ebecomes ready at terminali⁰ofc⁰. At most one link can be connected with a terminal.

When several ACs are connected by links, the resulting network is anactive system (AS) [18,19]. A state of an AS is a pair (S, E), whereS is the array of states of the components in the AS andEis the array of (possibly empty) events that are ready at the input terminals of the components. A state of the AS isquiescentiff all elements in Eareε(no event is ready). AtrajectoryTof an AS is a finite sequence of transitions of ACs in the AS, namelyT = [t1(c1), . . . , tq(cq)], which moves the AS from aninitial quiescent state to afinalquiescent state. In an AS, a terminal that is not connected with any link isdangling.

Anactive unit (AU) is an AS extended by a set of input terminals, a set of output terminals, and a set ofemergent events, where the input terminals are the dangling input terminals of the AS. Each emergent event is a pair(e(o), r), whereeis an event generated at the output terminaloof the AU andris a regular expression whose alphabet is a subset of the transitions of the ACs involved in the AU. The evente(o)emergesfrom a trajectory⁴of the AU when a sequence of transitions in the trajectory matchesr. The state of recognition of an emergent event(e(o), r)is maintained by amatcher, namely a deterministic finite automaton (DFA), derived fromr, in which each final state corresponds to the occurrence ofe(o). Remarkably, the notion of a matcher ofrdiffers from that of a recognizer ofr, as illustrated below.

Example 1. Let(e(o), r)be an emergent event in an AUU, withΣ={t1, t2, t3}being the alphabet of the regular expression

r=t1t⁺₂ t3|t2t⁺₃ t1. (1) Depicted on the left side of Fig. 1 is therecognizerofr, a DFA with the final state being marked by the emergent evente(o). When the final state5is reached, the emergent event eis generated at the output terminaloofU. Assume the following trajectoryT inU^∗,

T= [t₃,

T⁰

t₁, t₂, t₃ t₁, , t₁

T⁰⁰

, t₃]. (2)

T includes two overlapping subtrajectories matchingr, namelyT⁰ = [t₁, t₂, t₃]and T⁰⁰= [t₂, t₃, t₁], where the suffix[t₂, t₃]ofT⁰is a prefix ofT⁰⁰. Hence, the emergent evente(o)is expected to occur twice inT, in correspondence of the last transition ofT⁰ andT⁰⁰, respectively.

Assume further to trace the occurrences ofe(o)based on the recognizer ofrdis- played on the left side of Fig. 1. The sequence of states of this recognizer is outlined in the second row of Table 1, where each state is reached by the corresponding transition

3Transitions triggered by empty events are typically used for modeling state changes caused by external events (e.g. a lightning may cause the reaction of a sensor in a protection system).

4The notion of a trajectory defined for an AS is also valid for the AU incorporating the AS.

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Figure 1.The recognizer ofr =t1t⁺₂ t3|t2t⁺₃ t1, whereΣ ={t1, t2, t3}(left); the NFA obtained by the insertion ofε-transitions (center); and the matcherµ(e(o), r)(right).

ofTlisted in the first row of the table. As indicated in the third row, the emergent event e(o)is correctly generated at the fourth transition inT, which is the last transition of the subtrajectoryT⁰. At this point, since no transition exits the final state5, the recognizer starts again from its initial state0in order to keep matching. It first changes state to1 in correspondence oft₁, and witht₃(mismatch) it returns to0. The result is that, owing to the overlapping ofT⁰andT⁰⁰, the seconde(o)is not produced.

Given the pair(e(o), r), in order to recognize all (possibly overlapping) instances ofr, we need to transform the recognizer ofrinto thematcherofras follows:

1. An ε-transition is inserted into the recognizer from each non-initial state to the initial state;

2. The nondeterministic finite automaton (NFA) obtained in step 1 is determinized into an equivalent DFA;

3. The DFA generated in step 2 is minimized, thereby obtaining the matcher ofr.

The final states of the minimized DFA are marked by the emergent evente(o).

Example 2. With reference to Example 1, consider the recognizer of the regular expres- sionrdisplayed on the left side of Fig. 1. Outlined on the center is the NFA obtained by inserting fiveε-transitions (dashed arrows) toward the initial state (step 1). The DFA resulting from the determinization of the NFA is displayed on the right side of the figure (step 2). Incidentally, this DFA is also minimal, thus minimization (step 3) is not ap- plied. In conclusion, the DFA on the right side of Fig. 1 is the matcherµ(e(o), r), with

Table 1.Generation of emergent events with overlapping.

Transitions inT t3 t1 t2 t3 t1 t3

States of the recognizer ofr 0 1 3 5 1 0

Emergent events e(o)

States of the matcherµ(e(o), r)m0m1m3 m5 m6 m0

Emergent events e(o)e(o)

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the final statesm₅andm₆being marked bye(o). The dynamics of the matching performed byµ(e(o), r)on the trajectoryTis outlined in the last two rows of Table 1. In sharp contrast with the matching performed by the recognizer ofr, which produces only onee(o), the matcher correctly generates the emergent event twice, based on the two overlapping subtrajectories ofT, namelyT⁰= [t1, t2, t3]andT⁰⁰= [t2, t3, t1]. Unlike the recognizer ofr, after reaching the statem5and generatinge(o), the next transition t1moves the matcher to the other final statem6, thereby generating the second occurrence ofe(o)correctly.

A CASX is a directed tree, where each node is an AU and each arc(U,U⁰), withU being a child ofU⁰in the tree, is a set of links connecting all the output terminals ofU with some input terminals ofU⁰. A component / link is inXiff it is in an AU ofX.

Example 3. Outlined on the left side of Fig. 2 is a CAS calledX¯, involving the AUs A,B, andC, whereAhas one input terminal,Bhas one input terminal and one output terminal, andC has one output terminal. Since each AU has at most one child, the hierarchy ofX¯is linear (no branches). To avoid irrelevant details, the internal structure of the AUs is omitted.

A state ofX is a triple(S, E, M), whereS is the array of states of the ACs inX,E is the array of (possibly empty) events ready at the input terminals of the ACs inX, andMis the array of states of the matchers of the events emerging from all AUs inX. Like for ASs, a state ofX isquiescentiff all elements inEareε. AtrajectoryofX is a finite sequence of transitions of the ACs inX,T = [t1(c1), . . . , tq(cq)], which moves X from aninitialquiescent state to afinalquiescent state. Given an initial statex0of X, thespaceofX is a DFA

X^∗= (Σ, X, τ, x0, Xf), (3) whereΣis the alphabet, namely the set of transitions of the ACs inX,X is the set of states,τ is the transition function, namelyτ : X×Σ 7→ X, andXf ⊆X is the set of final states, which are quiescent. In other words, the sequence of symbols inΣ (transitions of ACs) marking a path (sequence of transitions) inX^∗fromx0to a final state is a trajectory ofX. Hence,X^∗is a finite representation of the (possibly infinite) set of trajectories ofX starting atx₀.

Within a trajectory of a CASX, each transition is eitherobservableorunobserv- able. The mode in which an observable transition is perceived is defined by theviewer V ofX, namely a set of pairs(t(c), `), wheret(c)is a transition of an ACc and`is anobservable label. Given a transitiont(c), if(t(c), `)∈ V, thent(c)is observable via label`; otherwise,t(c)is unobservable.

Assuming thatX includesnAUs, namelyU1, . . . ,Un, thetemporal observationof a trajectoryT ofX based on a viewerV, denotedT_V, is an array(O1, . . . , On)where,

∀i ∈[1.. n],Oiis theobservationofUi, defined as the sequence of observable labels associated with the observable transitions inT that are relevant to the ACs inUionly:

O_i= [`|t(c)∈T, c∈ Ui,(t(c), `)∈ V]. (4) In other words,T_V is the result of grouping by AUs the observable labels associated with the observable transitions inT.

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Not only each transition in a trajectoryT is either observable or unobservable; it also is either normalor faulty. Faulty transitions are defined by therulerRofX, a set of pairs(t(c), f), wheret(c)is a transition of an ACc andf is afault. Given a transitiont(c), if(t(c), f)∈ R, thent(c)is faulty via faultf; otherwise,t(c)is normal.

Thediagnosisof a trajectoryTofX based onR, denotedT_R, is defined as follows⁵: T_R={f |t(c)∈T,(t(c), f)∈ R }. (5) IfT_R 6= ∅, thenT is faulty; otherwise,T is normal. Even ifX^∗ includes an infinite number of trajectories, the domain of the possible diagnoses is always finite, this being the powerset2^F, whereFis the domain of faults involved inR.

3 Diagnosis Problem

WhenX performs an (unknown) trajectoryT, what is observed is a temporal observa- tionO=TV, whereVis the viewer ofX. However, owing to partial unobservability, several (possibly an infinite number of) trajectories may be compatible withO. Hence, several (a finite number of)candidate diagnosesmay be compatible withO. The goal is finding all these candidates. Adiagnosis problem℘(X,O)is an association between Xand a temporal observationO. Given the viewerVand the rulerRofX, thesolution to℘(X,O)is the set of candidate diagnoses

∆(℘(X,O)) ={T_R|T ∈ X^∗, T_V =O }. (6) Example 4. Consider the CASX¯in Example 3. Assume that the observable labels involved inV¯area, b, c, d, e, andf; the faults involved in the rulerR¯ arep, q, v, w, x, y, andz; the temporal observation is O¯ = ( ¯OA,O¯B,O¯C), whereO¯A = [e, f],O¯B = [c, d], andO¯C= [a, b, a]. We have that℘( ¯X,O)¯ is a diagnosis problem.

It should be clear that eqn. (6) is only a definition formalizing what the solution to a diagnosis problem is. It makes no sense to enumerate the candidate diagnoses as suggested by this definition, as the size ofX^∗is exponential in the number of ACs and input terminals. The spaceX^∗plays only a formal role and is never materialized. Even the reconstruction of the subspace ofX^∗that is compatible withOis out of the question because, in the worst case, it suffers from the same exponential complexity ofX^∗. So, what to do? The short answer is: focusing on the AUs rather than on the whole ofX. The important points are soundness and completeness: the set of diagnoses determined must coincide with the set of candidate diagnoses defined in eqn. (6).

4 Knowledge Compilation

The diagnosis process involves several tasks, which are performed eitherofflineoron- line, that is,beforeor while the CAS is being operated, respectively. The tasks performed offline are collectively called “knowledge compilation”, and the resulting data

5More generally, any set of faults is called a diagnosis.

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Figure 2.CASX¯(left) and unit spacesC^∗,B^∗, andA^∗.

structures, “compiled knowledge”. Compiled knowledge is meant to speed up the task performed online by the diagnosis engine. In offline processing, the AUs are processed independently from one another. For each AUU, three sorts of data structures are compiled in cascade: theunit spaceU^∗, theunit labelingU_δ, and theunit knowledgeU_∆, of which the latter is exploited online. Online processing depends on a diagnosis problem to be solved, specifically, on the observation, whereas offline processing does not.

Definition 1 (Unit space) LetUbe an AU involving an ASA. ThespaceofUis a DFA U^∗= (Σ, U, τ, u₀, U_f), (7) where: the alphabet Σ is the set of transitions of ACs in U; U is the set of states (S, E, M), where (S, E) is a state ofA and M is the array of states of the matchers of the emergent events ofU;u0= (S0, E0, M0)is the initial state, where(S0, E0) is a quiescent state ofAandM0is the array of initial states of the matchers;Uf ⊆U is the set of final states, where(S, E, M)∈Ufiff(S, E)is a quiescent state ofA; and τ :U×Σ7→U is the transition function, where(S, E, M)−−→^t(c) (S⁰, E⁰, M⁰)∈τ iff (S, E)−−→^t(c) (S⁰, E⁰)occurs inA,M⁰is the result of updatingM based ont(c), and (S⁰, E⁰, M⁰)is connected to a final state⁶.

Example 5. We assume that the space of each AU has been generated already, as shown next to the CASX¯in Fig. 2, namelyC^∗(left),B^∗(center), andA^∗(right). In each unit space, the states are identified by numbers, the final states are double circled, and the transitions are marked by relevant information only, namely: observable label (in blue), fault (in red), occurrence of an emergent event (prefixed by the “+” plus sign), and consumption of an event emerged from a child unit (prefixed by the “−” minus sign).

For instance, inB^∗, the transition from3 to4 is observable via the labeld, is faulty via the faultv, and generates the emergent eventβ. The transition from2to3, which is unobservable and normal, consumes the eventαemerging fromC, the child ofB.

Since the space of an AU does not depend on other AUs, the occurrence of a transition depending on an eventeemerging from a child AU in the CAS is not constrained by

6The requirement on connection means that there is a contiguous sequence of transitions from (S⁰, E⁰, M⁰)to a final state inUf.

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Figure 3.Unit labelingsCδ(left),Bδ(center), andAδ(right).

the actual readiness ofeat one input terminal, as this information is outside the scope of the AU. Yet, the enforcement of thisinterfaceconstraint is not neglected, but only deferred to the diagnosis engine operating online (cf. Section 5).

Definition 2 (Unit labeling) LetU^∗ = (Σ, U, τ, u0, Uf)be the space of an AUU in a CASX, letRbe the ruler ofX, and letδbe the domain of diagnoses inU. Thelabeling ofUis a DFA

Uδ= (Σ, U⁰, τ⁰, u⁰₀, U_f⁰), (8) where:U⁰ ⊆U ×δis the set of states;u⁰₀ = (u₀,∅)is the initial state;U_f⁰ ⊆U⁰ is the set of final states, with(u, δ)∈U_f⁰ifu∈U_f;τ⁰ : U⁰×Σ 7→U⁰is the transition function, with(u, δ)−−→^t(c) (u⁰, δ⁰)∈τ⁰iffu−−→^t(c) u⁰∈τand

δ⁰=

δ∪ {f}if(t(c), f)∈ R δ otherwise.

Example 6. Displayed on Fig. 3 are the unit labelingsC_δ,B_δ, andA_δ derived from the unit spaces outlined in Fig. 2. To facilitate subsequent referencing, states are re- identified by numbers. Owing to the additional field δ (set of faults), the number of states inU_δis generally larger than the number of states inU^∗.

Definition 3 (Unit knowledge) Let U be an AU in a CAS X, let Uδ be a labeling ofU, and let Rbe the ruler of X. Let U_δ⁰ be the NFA obtained from Uδ by substi- tuting the alphabet symbols marking the transitions as follows. For each transition (u, δ)−−→^t(c) (u⁰, δ⁰)inU_δ,t(c)is replaced with a triple(`, e, E), where: if(t(c), `⁰)∈ V, then` =`⁰, else` =ε; ift(c)consumes an evente⁰ emerging from a child unit, then e=e⁰, elsee=ε;Eis the (possibly empty) set of emergent events generated att(c) byU. Eventually, all triples(ε, ε,∅)are replaced byε. Theunit knowledgeU∆is the DFA obtained by the determinization ofU_δ⁰, where each final statesf ofU∆is marked

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Figure 4.Unit knowledgeC∆(left),B∆(center), andA∆(right).

by the aggregation of the diagnosis sets associated with the final states ofU_δ⁰ withinsf, denoted∆(sf)⁷.

Example 7. Shown in Fig. 4 are the unit knowledgeC_∆,B_∆, andA_∆, derived from the unit labelings displayed in Fig. 3, whereεelements in triples(`, α, β)are omitted and states are re-identified by numbers. For instance, consider the final state4ofC∆

(left of Fig. 4), which includes the states7 = (1,{x, z}),10 = (3,{x, z}), and14 = (2,{x, y, z})ofCδ. Since the state10 = (3,{x, z})inCδis final (it is final inC_δ⁰ too), the state4ofC∆is marked by{{x, z}}.

5 Diagnosis Engine

A diagnosis problem forX is solved online by thediagnosis engineby exploiting the knowledge of the AUs compiled offline. Each unit knowledgeU∆is constrained by the observation ofU and by the events emerging from the child AUs ofU.

Definition 4 (Abduction of leaf AU) Let U be an AU that is a leaf of the tree of a CAS, letU∆= (Σ, S, τ, s₀, S_f)be the knowledge ofU, and letO= [`₁, . . . , `_n]be an observation ofU. TheabductionofU is a DFA

U_O = (Σ, S⁰, τ⁰, s⁰₀, S_f⁰),

where:S⁰ ⊆ S ×[0.. n] is the set of states⁸; s⁰₀ = (s₀,0) is the initial state;S_f⁰ ⊆ S⁰ is the set of final states, where s⁰ ∈ S⁰_f iff s⁰ = (s, n), s ∈ S_f, with s⁰ being marked by ∆(s⁰) = ∆(s);τ⁰ : S⁰ ×Σ 7→ S⁰ is the transition function, where (s, j)−−−−→^(`,ε,E) (s⁰, j⁰)∈τ⁰iff(s⁰, j⁰)is connected to a final state,s−−−−→^(`,ε,E) s⁰∈τ, and

j⁰ =

j+ 1if`6=εand`j+1=`

j if`=ε. (9)

7Based on the algorithmSubset Construction[10], when an NFA is determinized into an equivalent DFA, each state in the DFA is identified by a subset of the states of the NFA.

8Each natural number in[0.. n]is called anindexofO.

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Figure 5.Unit abductionsCO(left),BO(center), andAO(right).

Example 8. Consider the unit knowledgeC∆displayed on the left side of Fig. 4. Let O¯C = [a, b, a]be the observation ofC. The unit abductionC_O is shown on the left side of Fig. 5.

To extend Def. 4 to nonleaf AUs, we introduce the notion of a unit interface in Def. 5 (generalized in Def. 8).

Definition 5 (Interface of leaf AU) LetU_Obe an abduction whereUis a leaf AU. Let U_O⁰ be the NFA obtained fromU_Oby substitutingEfor each transition symbol(`, ε, E) such thatE6=∅, andεfor all other symbols. TheinterfaceU=ofUis the DFA obtained by determinization ofU_O⁰ , where each final states⁰_fis marked by the union of the sets of diagnoses marking the final states ofU_O⁰ included ins⁰_f, denoted∆(s⁰_f).

Example 9. Considering the unit abductionC_O displayed on the left side of Fig. 5, the unit interfaceC₌is shown on the left side of Fig. 6, where all three states are final.

To extend the notion of a unit abduction to nonleaf AUs (Def. 7), we make use of the join operator defined below.

Definition 6 (Join) Thejoin “⊗” of two sets of diagnoses, ∆1 and∆2, is the set of diagnoses defined as follows:

∆1⊗∆2={δ|δ=δ1∪δ2, δ1∈∆1, δ2∈∆2}. (10) Definition 7 (Abduction of nonleaf AU) LetU be a nonleaf AU inX, letU1, . . . ,Uk

be the child AUs ofU inX, let U∆ = (Σ, S, τ, s₀, S_f) be the knowledge of U, let O = [`₁, . . . , `_n] be an observation ofU, letE be the domain of tuples of (possibly empty) events emerging from child AUs ready at the input terminals ofU, letU_i= = (Σ_i, S_i, τ_i, s_0i, S_fi)be the interface ofU_i,i∈[1.. k], and letS=S₁× · · · ×S_k. The abductionofU is a DFA

UO = (Σ⁰, S⁰, τ⁰, s⁰₀, S_f⁰),

where: Σ⁰ = Σ ∪Σ1∪ · · · ∪Σk; S⁰ ⊆ S ×S×E×[0.. n] is the set of states;

s⁰₀ = (s0,(s₀₁, . . . , s0k),(ε, . . . , ε),0)is the initial state;S_f⁰ ⊆ S⁰ is the set of final

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states, wheres⁰_f ∈S⁰_fiffs⁰_f = (s_f,(s_{f 1}, . . . , s_fk),(ε, . . . , ε), n),s_f ∈S_f,∀i ∈[1.. k], s_fi ∈S_fi, ands⁰_fis marked by the set of diagnoses

∆(s⁰_f) =∆(sf)⊗∆(s_{f 1})⊗ · · · ⊗∆(sfk); (11) τ⁰ :Σ⁰×S⁰7→S⁰is the transition function, where

(s,(s1, . . . , sk), E, j)−→^σ (s⁰,(s⁰₁, . . . , s⁰_k), E⁰, j⁰)∈τ⁰

iff(s⁰,(s⁰₁, . . . , s⁰_k), E⁰, j⁰)is connected to a final state and either of the following con- ditions holds:

(a) s−→^σ s⁰∈τ,σ= (`, e,E), either¯ e=εoreis ready inE,E⁰equalsEexcept that eis removed, and the indexj⁰is defined as in eqn. (9);

(b) s_i−→^σ s⁰_i ∈ τ_i,σ = E_i, all elements inE corresponding to the input emergent events inE_iareε,E⁰equalsEexcept that all events inE_iare inserted intoE⁰. Example 10. Consider the unit knowledge B_∆ displayed on the center of Fig. 4 and the unit interface C₌ displayed on the left side of Fig. 6. Let O¯B = [c, d] be the observation ofB. The unit abductionB_O is shown on the center of Fig. 5. Each state inB_O is marked by a quadruple(sB, sC, e, j), where sB is a state of B∆,sC is a state ofC₌(Cis the unique child ofB),eis the event emerging fromCand possibly ready (ife6= ε) at the input terminal ofB(B includes one input terminal only), and j is an index of the observationO¯B. Let∆¯be the set of diagnoses marking the final state8 = (8,2, ε,2). Based on eqn. (11),∆¯is generated via join∆(8)⊗∆(2), where

∆(8)is the set associated with the state8of the unit knowledgeB∆ (shown on the center of Fig. 4), namely {{v, w}}, and ∆(2) is the set associated with the state2 of the unit interfaceC₌(shown on the left side of Fig. 6), namely{{y, z}}. Hence,

∆¯={{v, w}} ⊗ {{y, z}}={{v, w, y, z}}. The unit interfaceB=, drawn fromBO, is displayed on the right side of Fig. 6.

Definition 8 (Interface of AU) LetUO be an abduction. LetU_O⁰ be the NFA obtained fromUO by substitutingEfor each transition symbol(`, e, E)such thatE 6=∅, andε for all other symbols. TheinterfaceU=ofU is the DFA obtained by determinization of U_O⁰ , where each final states⁰_f is marked by the union of the sets of diagnoses marking the final states ofU_O⁰ included ins⁰_f, denoted∆(s⁰_f).

Example 11. Considering the unit abductionB_O displayed on the center of Fig. 5, the unit interfaceB₌is shown on the right side of Fig. 6.

6 Soundness and Completeness

Once the abduction process reaches the rootU of the CASX, thereby generatingU_O, the solution to the diagnosis problem ℘(X,O) can be determined by collecting the diagnoses marking the final states ofU_O(Proposition 1).

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Figure 6.Unit interfacesC=(left) andB=(right).

Proposition 1 (Correctness) Let℘(X,O)be a diagnosis problem, letU be the AU at the root ofX, letU_O be the abduction ofU, withS_fbeing the set of final states, and let

∆(UO) = [

sf∈Sf

∆(sf). (12)

We have∆(℘(X,O)) =∆(UO).

To prove Proposition 1, further terminology is required. LetU¯ be the CAS withinX rooted inU. Therestrictionof℘(X,O)onU¯is a diagnosis problem ℘( ¯U,O)¯ where the viewerV¯ ofU¯ is the restriction of the viewer of X on pairs(t(c), `)such thatc is a component in U¯, the rulerR¯ of U¯ is the restriction of the ruler ofX on pairs (t(c), f)such thatcis a component inU¯, the initial state ofU¯ is the restriction of the initial state ofX on the components and links inU¯, andO¯ is the restriction ofOon observations of AUs inU¯. The notion of a trajectory is extended to any DFA involved in the diagnosis task, namely unit labeling, unit knowledge, unit abduction, and unit interface. Atrajectoryin any such DFA is a string of the regular language of the DFA, that is, the sequence of symbols in the alphabet which mark the sequence of transitions from the initial state to a final state of the DFA; such a final state is called theaccepting stateof the trajectory. The set of trajectories in a DFAD(the regular language ofD) is denoted bykDk. For instance,T ∈ kU_Okdenotes a trajectoryT in the abductionU_O.

Apathpin a DFADis the sequence of transitions yielding a trajectory[σ₁, . . . , σ_n] inD, namelyp=s₀ −→^σ¹ s₁−→ · · ·^σ² −^σ−→ⁿ s_n. AsemipathinDis a prefix of a path in D. Asubpathp⁰inDis a contiguous sequence of transitions within a path, from state si to statesj, namelysi

σ_i+1

−−−→s1+1 σ_i+2

−−−→ · · ·−−−→^σ^j−1 s_j−1 −→^σ^j sj, concisely denoted s_i=⇒^σ s_j, whereσ= [σ_i+1, . . . , σ_j−1, σ_j]is thesubtrajectorygenerated byp⁰.

The notion of an interface is extended to any trajectory. Theinterfaceof a trajectory T, denoted=(T), is the sequence of nonempty sets of emergent events occurring inT. Specifically, ifTis a trajectory inU₌, then=(T) =T. IfT is a trajectory in eitherU_O orU∆, then=(T) = [E|(`, e, E)∈T, E6=∅]. IfT is a trajectory in eitherU^∗orUδ, then each setEin=(T)is the nonempty set of events emerging at the occurrence of a component transition inT.

LetG= (N, A)be a directed acyclic graph (DAG), whereNis the set of nodes and Ais the set of arcs. Letnandn⁰be two nodes inN. We say thatnprecedesn⁰, denoted n≺n⁰, iffn⁰is reachable fromnby a contiguous sequence of arcs. Atopological sort S inGis a sequence of all nodes inN (with each node occurring once) such that, for

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each pair(n, n⁰)of nodes inS, ifn≺n⁰inG, thenn≺n⁰inS. The whole (finite) set of topological sorts inGis denotedkGk.

Based on the terminology introduced above, a sketch of the proof of Proposition 1 can be given on the ground of Lemma 11, Lemma 12, Corollary 11, and Lemma 13.

Lemma 11 (Mapping) Let D = (Σ, S, τ, s₀, S_f) be a DFA. LetΣ⁰ be an alphabet (possibly includingε) and letΣ7→Σ⁰be a surjective mapping fromΣtoΣ⁰. LetN = (Σ⁰, S, τ⁰, s₀, S_f)be the NFA derived fromDby replacing each transitions−→^σ σ∈τ withs ^σ

0

−→σ∈τ⁰, whereσ⁰is the symbol inΣ⁰mapping the symbolσinΣ. LetD⁰be the DFA obtained by determinization ofN. IfT⁰is a trajectory inkD⁰kwith accepting states⁰_f, then, for each final statesf∈s⁰_f, there is a trajectoryT inkDkwith accepting statesfsuch thatT⁰is the mapping ofTbased onΣ7→Σ⁰.

Proof. By induction onT⁰.

(Basis) According to theSubset Constructiondeterminization algorithm, ifs∈s⁰₀, wheres⁰₀ is the initial state of D⁰, then there is a paths0

ε^∗

=⇒s inN whereε^∗ is a (possibly empty) sequence ofε. Hence, there is a paths0

=⇒σ sinDsuch thatε^∗is the mapping ofσbased onΣ7→Σ⁰.

(Induction) Assume that there is a semipaths⁰₀ ^σ

0

=⇒s⁰inD⁰such that for eachs∈s⁰ there is a semipath s0

=σ

⇒sinDwhereσ⁰ is the mapping ofσ based on Σ 7→ Σ⁰. Consider the new semipaths⁰₀ ^σ

0

=⇒s⁰ −→^¯^σ ¯s⁰ inD⁰. According toSubset Construction, for each state¯s∈¯s⁰there is a states∈s⁰such thats=^σ^¯⇒ⁿ s¯is a subpath inN whereσ¯n

starts withσ, followed by a (possibly empty) sequence of¯ ε. Hence, there is a subpath s=^¯^σ⇒^d s¯inD such that σ¯d maps to [¯σ]. Thus, by virtue of the induction hypothesis,

σ∪σ¯dmaps toσ⁰∪[¯σ]based onΣ7→Σ⁰.

Corollary 11 With reference to the assumptions stated in Lemma 11, ifT⁰is generated by a pathp⁰ =s⁰₀ ^σ

0

−→1 s⁰₁ ^σ

0

−→2 s⁰₂ ^σ

0

−→ · · ·3 ^σ 0

−−−→n−1 s⁰_n−1 ^σ

0

−−→n s⁰_n inD⁰, then there is a pathp= s₀ =^σ⇒¹ s₁ =^σ⇒² s₂ =^σ⇒ · · ·³ ===^σⁿ⁻¹⇒ s_n−1 =^σ⇒ⁿ s_n inDsuch that,∀i ∈[1.. n], s_i−1∈s⁰_i−1,s_i ∈s⁰_i, and[σ_i⁰]is the mapping ofσ_ibased onΣ7→Σ⁰.

Lemma 12 (Soundness) Ifs⁰_f is a final state inUO,δ ∈ ∆(s⁰_f), andT ∈ kUOkwith accepting states⁰_f, thenδ∈∆(℘( ¯U,O), where¯ δ= ¯TR¯,T¯∈ kU¯^∗k, and=( ¯T) ==(T).

Proof. By bottom-up induction on the tree ofU¯.

(Basis) Assume thatU is a leaf AU. Sinces⁰_f = (sf, n)is a final state inUO,δ ∈

∆(s⁰_f), andT ∈ kUOkwith accepting states⁰_f, based on Def. 4 and Lemma 11, there is T_∆ ∈ kU∆kwith accepting states_f such thatδ∈ ∆(s_f)and=(T∆) ==(T). Hence, based on Def. 3 and Lemma 11, there isT_δ∈ kUδkwith accepting state(s, δ)such that

=(Tδ) = =(T∆) = =(T). Therefore, based on Def. 2, there isT¯ ∈ kU^∗ksuch that T¯V¯ = ¯OandT¯R¯ = δ; in other words, according to eqn. (6),δ ∈ ∆(℘( ¯U,O). Also,¯

=( ¯T) ==(T_δ) ==(T).

(Induction) Assume thatUis the parent of the AUsU₁, . . . ,U_k, where∀i∈[1.. k], ifs⁰_ifis a final state inUiO,δi ∈∆(s⁰_if), andTi ∈ kUiOkwith accepting states⁰_if, then δi∈∆(℘( ¯Ui,O¯i), whereδi= ¯TiR¯i,T¯i∈ kU¯_i^∗k, and=( ¯Ti) ==(Ti).

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Sinces⁰_f = (s_f,(s_1f, . . . , s_kf),(ε, . . . , ε), n)is a final state inU_O,δ ∈∆(s⁰_f), and T ∈ kU_Ok with accepting state s⁰_f, according to eqn. (11), δ ∈ ∆(s⁰_f) = ∆(s_f)⊗

∆(s_1f)⊗ · · · ⊗∆(s_kf), where,∀i∈[1.. k],s_ifis final inU_i=. Hence, based on Def. 6, δ = δ0∪δ1∪ · · · ∪δk, whereδ0 ∈ ∆(sf) and, ∀i ∈ [1.. k],δi ∈ ∆(sif). Also, based on Def. 8 and Lemma 11,∀i∈[1.. k], there isTi ∈ kUiOkwith accepting state s⁰_if ∈ sif such thatδi ∈ s⁰_if and all emergent events inTi are consumed inT. Thus, by virtue of the induction hypothesis, we haveδi ∈ ∆(℘( ¯Ui,O¯i), whereδi = ¯TiR¯i, T¯i ∈ kU¯_i^∗k, and =( ¯Ti) = =(Ti). According to Def. 7, the trajectory T is composed of triples (`, e, E) interspersed by elements Ei, with eachEi being a nonempty set of events emerging fromUi. Note that, for each Ei inT there is one transition inT¯i

generatingEi. Thus, eachT¯iis composed of transitionst(c), wherecis a component inU¯i, in which there are some transitions generating setsEi of emergent events. The sequence ofEi generated inT¯i equals the subsequence ofEi inT,i ∈ [1.. k]. So, there is a sequential correspondence between eachE_i inT and each transition inT¯_i generating the set E_i of emergent events. According to Def. 7 and Corollary 11, if s⁰₀ ^σ

0

=⇒1 s⁰₁ −−→^Eⁱ¹ s⁰⁰₁ ^σ

0

=⇒2 s⁰₂ −−→^Eⁱ² s⁰⁰₂ ^σ

0

=⇒ · · ·3 ^σ 0

=⇒n s⁰_n is the path inUO generating the trajectoryT =σ⁰₁∪[Ei1]∪σ₂⁰∪[E_i2]∪σ⁰₃∪· · ·∪σ_n⁰, then there is a paths₀=^σ⇒¹ s₁=^σ⇒² s₂=^σ⇒ · · ·³ =^σ⇒ⁿ s_ninUδgenerating the trajectoryT^∗=σ₁∪σ2∪· · ·∪σn,T^∗∈ kU^∗k, such thatT^∗ generates the observation ofU and the diagnosisδ₀. Now, consider the DAGGconstructed by the following four steps, where each node is either a component transitions or a setE_iof emergent events, while arcs indicate precedence dependencies between these nodes. Step 1: in T, substituteσj for each σ⁰_j,j ∈ [1.. n]; this step substitutes sequences of transitions of components inUfor corresponding sequences of triples(`, e, E)inT; Step 2: transform each trajectoryT¯i= [t1, t2, . . . , tn_i],i∈[1.. k], into a connected sequence of nodest1→t2 → · · · →tn_i, where arrows indicate arcs (precedence dependencies); Step 3: transformT into a connected sequence of nodes in the same way as in step 2; Step 4: for each transitiontiinT¯i generating the setEi

of emergent events, insert an arc from ti to the correspondingEi inT. This results in a DAG, namely adependency graphG, where eachEiis entered by an arc from a transition inT¯i,i∈[1.. k]. LetT¯be the sequence of transitions relevant to a topological sort ofG, where allEiare eventually removed. As such,T¯is a sequence of transitions of components inU¯fulfilling the precedences imposed byG. Remarkably,T¯fulfills the following properties:(1) ¯T ∈ kU¯^∗k, because the component transitions in eachT¯_iare only constrained by the emptiness of the output terminals ofU_i, while the component transitions inT are only constrained by the availability of the events emerging from U₁, . . . ,U_k, which are checked by the mode in which the abductionU_O is generated;

(2) ¯TV¯ = ¯O, because the sequence of observable labels generated by transitions inT equals the observation ofUand,∀i∈[1.. k],T¯iV¯i=Oi; and(3)the sequence of faults generated by the the transitions inT equalsδ0 and,∀i ∈ [1.. k],T¯iR¯i =δi. In other words,T¯R¯ =δ=δ0∪δ1∪ · · · ∪δk; hence,δ∈∆(℘( ¯U,O). Also,¯ =( ¯T) ==(T).

Lemma 13 (Completeness) Ifδ ∈ ∆(℘( ¯U,O), where¯ δ = ¯TR¯ andT¯ ∈ kU¯^∗k, then there isT ∈ kU_Okending ins⁰_fsuch thatδ∈∆(s⁰_f)and=(T) ==( ¯T).

Proof. By bottom-up induction on the tree ofU¯.

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(Basis) Assume thatU is a leaf AU. Sinceδ ∈ ∆(℘( ¯U,O), where¯ δ = ¯TR¯ and T¯ ∈ kU¯^∗k, based on Def. 2, there isT_δ ∈ kU_δk with accepting state(s, δ)such that

=(T_δ) = =( ¯T). Hence, based on Def. 3, there isT_∆ ∈ kU_∆kwith accepting states_f such thatδ ∈ ∆(sf)and=(T∆) = =(Tδ) = =( ¯T). Thus, based on Def. 4, there is T ∈ kU_Okending ins⁰_f= (sf, n)such thatδ∈∆(s⁰_f)and=(T) ==(T∆) ==( ¯T).

(Induction) Assume thatUis the parent of the AUsU1, . . . ,Uk, where,∀i∈[1.. k], ifδi∈∆(℘( ¯Ui,O¯i)), whereδi = ¯TiR¯iandT¯i∈ kU¯_i^∗k, then there isTi∈ kU_iOkwith accepting states⁰_ifsuch thatδi∈∆(s_if⁰ )and=(Ti) ==( ¯Ti).

Sinceδ∈∆(℘( ¯U,O), where¯ δ= ¯TR¯ andT¯ ∈ kU¯^∗k, we haveδ=δ0∪δ1∪ · · · ∪ δk, whereδ0includes the faults of the components in the AUU and,∀i ∈ [1.. k],δi

includes the faults of the components in the CASU¯i. Since eachδiis the diagnosis of the trajectoryT¯icorresponding to the restriction ofT¯on the transitions of the components in U¯i such that T¯i ∈ kU¯_i^∗k, T¯iV¯i = Oi, and T¯iR¯i = δi, based on eqn. (6), δi ∈

℘( ¯Ui,O¯i). Hence, by virtue of the induction hypothesis, there is T_i ∈ kU_iOk with accepting states⁰_if such thatδ_i ∈∆(s_if⁰ )and=(Ti) ==( ¯T_i). Also, based on Def. 8, there is T_i⁰ ∈ kU_i=k with accepting state s_if such thatδ_i ∈ ∆(s_if) and=(T_i⁰) =

=(T_i) = =( ¯T_i). Let T⁰ be the subtrajectory of T¯ including only the transitions of components inU. As such,T⁰ ∈ kU^∗k,T_V⁰ equals the observation inOrelevant toU, andT_R⁰ =δ0. Hence, based on Def. 2, there isTδ ∈ kUδkwith accepting state(s, δ0) such that=(Tδ) ==(T). Also, based on Def. 3, there isT∆ ∈ kU∆kwith accepting state sf such thatδ ∈ ∆(sf)and=(T∆) = =(Tδ) = =(T). Thus, based on Def. 7, there isT ∈ kU_Okwith accepting states⁰_f = (sf,(s_1f, . . . , skf),(ε, . . . , ε), n)such that δ0∈∆(sf)and,∀i∈[1.. k],δi∈∆(sif). Sinceδ=δ0∪δ1∪ · · · ∪δk, based on Def. 6 and according to eqn. (11),δ ∈∆(sf)⊗∆(s_1f)⊗ · · · ⊗∆(skf); that is,δ ∈ ∆(s⁰_f).

Also,=(T) ==( ¯T).

Example 12. Consider the unit knowledgeA∆displayed on the right side of Fig. 4 and the unit interface B= displayed on the right side of Fig. 6. LetO¯A = [e, f]be the observation ofA. The unit abductionAO is shown on the right side of Fig. 5. Let∆¯ be the set of diagnoses marking the final state5 = (5,1, ε,2). Based on eqn. (11),∆¯ is generated via join∆(5)⊗∆(1), where∆(5)is the set associated with the state5of the unit knowledgeA_∆(on the right side of Fig. 4), namely{{q}}, and∆(1)is the set associated with the state1of the unit interfaceB₌(on the right side of Fig. 6), namely {{v, w, y, z}}. Hence,∆¯={{q, v, w, y, z}}. Based on Proposition 1,∆¯is the solution to the diagnosis problem℘( ¯X,O)¯ defined in Example 4.

7 Computational Complexity

Had we adapted the diagnosis technique proposed for ASs [18,19] to the diagnosis of CASs, we would have inherited (and even exacerbated) its poor performance (see the experimental results in [17]). In contrast with diagnosis of ASs, the diagnosis engine described in Section 5 does not require the abduction of the whole system. Instead, it focuses on the abduction of each single AU based on the interface constraints coming from the child AUs. In doing so, it exploits the unit knowledge compiled offline.

We analyze the time complexity of solving a diagnosis problem℘(X,O)based on the (not unreasonable) assumption that the processing time is proportional to the size

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(number of states) of the data structures generated by the diagnosis engine. Further- more, we make the followingbounding assumptions:X is composed ofnAUs; each nonleaf AU hascchild AUs, hashinput terminals, and definesmemergent events; each unit knowledge (generated offline) includeskstates; the length of each AU observation in℘(X,O)iso. We also assume that the size of the DFA obtained by the determinization of an NFA compares to the size of the NFA⁹. We call this thedeterminization assumption. We first consider the (upper bound of the) complexityCof the abduction U_Oof a leaf AU (at level zero, that is, without children). Based on Def. 4,C_U_O =k·o.

In order to estimate the complexity of each unit abductionU_O at level one, where all children ofUare leaf AUs, two steps have to be analyzed:(1)generation ofcinterfaces and(2)generation ofUObased on Def. 7. As to step(1), on the ground of the determinization assumption, the number of states of the interfaces of the child AUs isc·k·o.

Based on Def. 7, the (upper bound of the) number of states generated by step(2)for each unit abduction at level one isk·(k·o)^c·m^h·o= (k·o)^c+1·m^h. Owing to the factor(k·o)^c+1, the contributionc·k·oof step(1)can be neglected; hence,

CUO = (k·o)^c+1·m^h. (13) The complexity of each unit abductionUO at the second level (whereU is the grand- parent of leaf AUs) can be computed as before, where the size of each interface equals CUOin eqn. (13), namely

CUO =k·((k·o)^c+1·m^h)^c·m^h·o= (k·o)^c²^+c+1·m^(c+1)·h. (14) At leveld(depth of the tree) of the root AU, the complexity of the unit abduction is

CUO = (k·o)^c^d^+c^d−1^+···+c²^+c+1·m^(c^d−1^+···+c²^+c+1)·h. (15) Since1 +c+c²+· · ·+c^d = n, withnbeing the number of AUs in the CAS, the dominant factor in eqn. (15) is (k·o)ⁿ. In other words, the complexity of the unit abduction is exponential in the number of AUs in the CAS.

Finally, we consider the space complexity of knowledge compilation, which is performed offline. We assume that each AU includes p components and l links, with each component havingsstates and generatingqdifferent events for each connected link. Each matcher (the DFA recognizing the occurrence of an emergent event) hasµ states. The number of possible faults isf. The space complexity of the unit spaceU^∗is C_U^∗=s^p·(q+ 1)^`·µ^m. The space complexity of the unit labelingU_δ is

CUδ =CU^∗·2^f =s^p·(q+ 1)^`·µ^m·2^f. (16) According to the determinization assumption, the complexity of the unit knowledge equals the complexity of the unit labeling, namelyCU∆=CUδ.

9In the worst case, ifnis the number of states of the NFA, then the number of states of the DFA is2ⁿ. However, this is an extremely improbable scenario since, in practice, the size of the DFA resulting from the determinization of an NFA generated randomly typically compares with the size of the NFA (cf. Fig. 4). If the NFA includes a considerable percentage ofε-transitions, then the size of the DFA is likely to be substantially smaller than the size of the NFA [4].