Inference-based Ambiguity Management in Decentralized Decision-Making: Decentralized Diagnosis of Discrete Event Systems

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1 Inerene-based Ambiguity Management in Deentralized Deision-Making: Deentralized Diagnosis o Disrete Event Systems Ratnesh Kumar and Shigemasa Takai Abstrat The task o deentralized deision-making involves interation o a set o loal deision-makers, eah o whih operates under limited sensing apabilities and is thus subjeted to ambiguity during the proess o deision-making. In a prior work [4] we made a key observation that suh ambiguities are o diering gradations and presented a ramework or inerening over various loal ontrol deisions o varying ambiguity levels to arrive at a global ontrol deision. We develop a similar ramework or perorming diagnosis in a deentralized setting. For eah event-trae exeuted by a system being monitored, eah loal diagnoser issues its own diagnosis deision (ailure or non-ailure or unsure, tagged with a ertain ambiguity level (zero being the minimum. A global diagnosis deision is taken to be a winning loal diagnosis deision, i.e., one with a minimum ambiguity level. The omputation o an ambiguity level or a loal deision requires an assessment o the sel-ambiguities as well as the ambiguities o the others, and an inerene based up on suh knowledge. In order to haraterize the lass o systems or whih any ault an be deteted within a uniormly bounded delay, we introdue the notion o N-inerene-diagnosability (or ailures, where the index N represents the maximum ambiguity level o any winning loal deision. We show that the odiagnosability introdued in [5] is the same as 0-inerenediagnosability; the onditional odiagnosability introdued in [13] is a type o 1-inerene-diagnosability; and the lass o higher-index inerene-diagnosable systems stritly subsumes the lass o lower-index ones I. INTRODUCTION In any deentralized deision-making paradigm, suh as deentralized ontrol or diagnosis, multiple deision-makers, eah with its limited sensing and/or ontrol apabilities, interat to ome up with the global deisions. Presene o limited sensing apabilities an lead to ambiguity in knowing the system state and thereby ambiguity in deision-making. Consider or example the problem o deentralized diagnosis [3], [9], [11], [1], [12], [5], [13] o disrete event systems (DESs. Suppose there exist two traes that are exeutable in the plant and are indistinguishable to a loal diagnoser, and one o the traes is a ailure trae while the other one is a non-ailure trae. Sine these two traes are indistinguishable, upon reeiving their observation, the loal diagnoser will be ambiguous about whether or not a ailure The researh was supported in part by the National Siene Foundation under the grants NSF-ECS , NSF-ECS , NSF-EPNES , and NSF-ECS , a DoD-EPSCoR grant through the Oie o Naval Researh under the grant N , and MEXT under Grant-in-Aid or Sientii Researh R. Kumar is with the Department o Eletrial and Computer Engineering, Iowa State University, Ames, Iowa , USA, rkumar@iastate.edu S. Takai is with the Department o Eletronis and Inormation Siene, Kyoto Institute o Tehnology, Sakyo-ku, Kyoto , Japan, takai@dj.kit.a.jp ourred. Similar situations an also arise in the setting o distributed diagnosis (i.e., one involving ommuniation among the loal diagnosers sine the problem o distributed diagnosis an be redued to an instane o deentralized diagnosis (i.e., one involving no ommuniation among the loal diagnosers [6]. Similarly, suh ambiguities an also be present in the setting o deentralized deision making or ontrol [4]. In the ontext o deentralized ontrol, a knowledge-based mehanism or assessing the sel-ambiguities was presented in [7], and later the same arhiteture was used or assessing the sel-ambiguities as well as the ambiguities o the others in [8]. The proess o utilizing the knowledge o the selambiguities together with the ambiguities o the others or the sake o deision-making was reerred to as inerening in [8] and onditioning in [14]. As is argued in [4], these prior inerening-based approahes were limited by a singlelevel o inerening, and a omprehensive ramework allowing multi-level inerening over various loal ontrol deisions o varying levels o ambiguity was irst presented in [4]. In the ontext o deentralized diagnosis, [5] suggested the ollowing simple tehnique or the management o ambiguity in the ontext o deentralized diagnosis. When a loal diagnoser is ambiguous about whether or not a ailure has ourred it simply opts to issue no diagnosis deision, i.e., a diagnosis deision is issued by a loal diagnoser only when it is unambiguous about it. This led to the introdution o the notion o odiagnosability in [5] that required that or eah ailure trae exeutable by the system being monitored, there be at least one loal diagnoser that an unambiguously determine this within a bounded number o additional transition-exeutions. An extension reported in [13] onsidered deentralized diagnosis based on the ideas o onditioning introdued in the setting o deentralized ontrol and introdued the notion o onditional odiagnosability that is weaker than odiagnosability. As is the ase with onditional oobservability, onditional odiagnosability involves a single level o inerening over the knowledge about ambiguities. In this paper we build up on the ideas o inerenebased ambiguity management in the setting o deentralized ontrol [4] and develop a ramework or inerene-based deentralized diagnosis. Our ramework supports (i inerening utilizing the knowledge o the sel-ambiguities as well as the ambiguities o the other deision makers, (ii inerening over an arbitrary number o levels o ambiguity. Eah loal diagnoser uses its observations o the system

2 behavior to ome up with its diagnosis deision together with a grade or level o ambiguity or that diagnosis deision. The omputation o an ambiguity level o a loal deision requires the assessment o the sel-ambiguities together with the ambiguities o the others. A diagnosis deision with level-zero ambiguity is taken when the loal diagnoser is unambiguous about its ailure (resp., non-ailure deision. This happens when all traes, produing the same observation as the one reeived, are ailure (resp., non-ailure traes. Otherwise, a higher ambiguity level diagnosis deision is issued. For example a ailure deision o level-one ambiguity is issued ollowing a ertain observation i there exist ertain traes, produing the same observation as the one reeived, suh that some o them are ailure traes while others are non-ailure traes. Existene o suh traes is learly a soure o ambiguity or the loal diagnoser. Yet, suppose the loal diagnoser is able to determine that or eah non-ailure trae, produing the same observation as the one reeived, there exists another loal diagnoser whih an issue a non-ailure diagnosis deision with level-zero ambiguity, then the loal diagnoser issues a ailure deision with level-one ambiguity. In general a loal diagnoser will issue a ailure (resp., nonailure deision with an ambiguity level N ollowing a ertain observation i or eah non-ailure (resp., ailure trae, produing the same observation as the one reeived, there exists another loal diagnoser that an issue a non-ailure (resp., ailure deision with an ambiguity level at most N 1. Clearly, a level-zero ambiguity diagnosis deision is based on assessment o only the sel-ambiguities, whereas a level-n ambiguity diagnosis deision is based on assessment o the sel-ambiguities together with the ambiguities o other loal diagnosers suh that or eah trae, that reates the ambiguity, there exists another loal diagnoser whih an issue a diagnosis deision with an ambiguity level at most N 1. Note in ertain situations it is possible that a loal diagnosis deision is neither ailure nor non-ailure, but unsure. As outlined above, our ramework allows inerening involving multiple-levels o ambiguities. Following the exeution o eah event all loal diagnosers reeiving a new observation issue a new diagnosis deision, tagged with a ertain level o ambiguity. The global diagnosis deision is taken to be the same as a loal diagnosis deision whose ambiguity level is the minimum. (Suh a loal deision an be onsidered to be a winning loal deision. We ormulate the notion o inerene-diagnosability to haraterize the lass o diagnosable systems in the proposed ramework o deentralized diagnosis. A system is said to be diagnosable i the global diagnosis deisions are suh that there are no missed detetions, i.e., diagnosis deision is ailure ollowing the exeution o any ailure trae within a bounded number o additional transition-exeutions, and there are no alse alarms, i.e., diagnosis deision ollowing a non-ailure (resp., ailure trae is not ailure (resp., non-ailure. For a diagnosable system, when the ambiguity level o all winning deisions are upper bounded by N, the system is said to be N-inerene-diagnosable. We study various properties o N-inerene diagnosability. In partiular we show that the notion o odiagnosability studied in [5] is the same as the notion o 0-inerene-diagnosability. Further the notion o onditional odiagnosability introdued in [13] is similar to that o 1-inerene-diagnosability. (The dierene omes beause o the use o an estimate in [13] that ignores the ourrene o unobservable events at the end. For this reason, 1-inerene-diagnosability is stronger than that the onditional odiagnosability. We also show an example that is 2-inerene-diagnosable but not onditionally diagnosable. Thus the ramework presented here subsumes and extends the ones reported earlier. We also establish that the lasses o N-inerene-diagnosable systems orm a monotonially inreasing sequene as a untion o N. We also provide an eetive test to veriy whether a given system is N- inerene-diagnosable. II. NOTATION AND PRELIMINARIES We onsider a DES modeled by a inite nondeterministi automaton G = (X, Σ, α, X 0, X m, where X is the inite set o states, Σ is the inite set o events, a partial untion α : X Σ {ε} 2 X is the transition untion, X 0 X is the set o initial states, and X m X is the set o marked or aepting states. G is said to be deterministi i the transition untion an be written as a partial untion α : X Σ X and X 0 = 1. Let Σ be the set o all inite sequenes o events inluding the empty sequene ε. Elements o Σ are alled traes, and subsets o Σ are alled languages. The transition untion α an be generalized to α : 2 X Σ 2 X in a natural way. The generated and marked (or aepted languages o G are respetively deined as, L(G := {s Σ α(x 0, s }, and L m (G := {s Σ α(x 0, s X m }. For a language K, the set o all preixes o traes in K is denoted by pr(k, i.e., pr(k = {s Σ t Σ ; st K}. K is said to be (preix-losed i K = pr(k. A language K is said to be deadlok-ree i or any s K, exists a trae t ε suh that st K; otherwise s is alled a deadloking trae o K. For eah trae s Σ, s denotes its length. For any m N, where N denotes the set o all nonnegative integers, let Σ m := {s Σ s m} and Σ m := {s Σ s m} denote the set o all traes with m or more events and m or less events respetively. III. INFERENCE-BASED DECENTRALIZED DIAGNOSIS FRAMEWORK In this paper, we onsider deentralized diagnosis where n loal diagnosers perorm the task o diagnosis without sharing their observations. We assume that the limited sensing apabilities o the ith loal diagnoser D i (i I := {1, 2,, n} an be represented as the loal observation mask, M i : Σ {ε} i {ε}, where i is the set o loally observed symbols, and M i (ε = ε. The notation Σ i,uo denotes the set o loally unobservable events, i.e., Σ i,uo = {σ Σ M i (σ = ε}.

3 Let L be a losed language representing a generated language o a plant (system to be diagnosed, and K L be a losed language representing a non-ailure speiiation language. Traes in L K are onsidered ailure traes and the task o diagnosis is to determine the exeution o any trae in L K within an additional bounded number o system exeutions. Without loss o generality, the plant language L an be taken to be deadlok-ree. Otherwise we an extend eah deadloking trae by an unbounded sequene o a newly added event that is unobservable to all diagnosers. This will make the language deadlok-ree without altering any diagnosability property sine the newly added event does not produe any observation to any o the diagnosers. Let the set C = {0, 1, φ} be the set o diagnosis deisions, where 0 represents a non-ailure deision, 1 represents a ailure deision, and φ represents an unsure deision. Eah inerene-based loal diagnoser D i is deined as a map D i : M i (L C N, where or eah s L, D i (M i (s = ( i (M i (s, n i (M i (s. Here i (M i (s C denotes the diagnosis deision o D i ollowing an observation M i (s M i (L, and n i (M i (s N denotes the ambiguity level o the diagnosis deision o D i. Let n(s be the minimal ambiguity level o loal deisions, i.e., n(s := min n i(m i (s. The deentralized diagnoser {D i } that onsists o loal diagnosers D i (i I issues global diagnosis deisions. Formally, {D i } is deined as a map {D i } : L C. For eah s L, the diagnosis deision {D i } (s is given as ollows: 0, i i I s.t. n i (M i (s = n(s; i (M i (s = 0 {D i } (s = 1, i i I s.t. n i (M i (s = n(s; i (M i (s = 1 φ, otherwise. In other words, the global diagnosis deision is taken to be the same as the minimum ambiguity level loal diagnosis deision. IV. EXISTENCE/SYNTHESIS OF INFERENCE-BASED DECENTRALIZED DIAGNOSERS Given a plant language L and a non-ailure speiiation language K L, we indutively deine a monotonially dereasing sequene o language pairs {(F k, H k } as ollows: Base step: Indution step: F 0 := L K, H 0 := K. F k+1 := F k H k+1 := H k i M i (H k i M i (F k,. The omputation o the sequene o languages (F k, H k starts with F 0 = L K, the set o Failure traes, and H 0 = K, the set o non-ailure or Healthy traes. Note that F k+1 is a sublanguage o F k onsisting o those traes or whih or eah i I there exists an M i -indistinguishable trae in H k. As a result when the plant exeutes a trae in F k+1 all the loal diagnosers will be ambiguous as to whether the exeuted trae is in F k+1 or in H k. The sublanguage H k+1 o H k an be understood in a similar ashion. Using the sequenes o language pairs (F k, H k, a loal diagnoser omputes its diagnosis deision and assoiates a level o ambiguity with suh a deision as ollows. Let N N be a given nonnegative integer. (N represents an index o diagnosability to be elaborated later. For eah s L, the ith loal diagnoser D i omputes n i (M i(s := min{n + 1, {k N M i (s / M i (H k }}, (1 n h i (M i (s := min{n + 1, {k N M i (s / M i (F k }}. (2 Note that n i (M i(s and n h i (M i(s are bounded above by N + 1. Here n i (M i(s represents the ambiguity level o a ailure deision ontemplated by the ith diagnoser ollowing the observation M i (s. When n i (M i(s < N +1, it denotes the minimum index k suh that the observation M i (s does not math with the observations o any o the traes in H k. Similarly, the notation n h i (M i(s represents the ambiguity level o a non-ailure deision ontemplated by the ith diagnoser ollowing the observation M i (s. Whih o the two ontemplated deisions is ultimately issued is deided by omparing the two ambiguity levels, n i (M i(s vs. n h i (M i(s, and avoring the smaller one. This is ormalized next. For a loal diagnoser D i : M i (L C N, its diagnosis deision and ambiguity level ollowing an observation M i (s M i (L, i.e., D i (M i (s = ( i (M i (s, n i (M i (s, is determined as ollows: and i (M i (s = 1, i n i (M i(s < n h i (M i(s 0, i n h i (M i(s < n i (M i(s φ, i n i (M i(s = n h i (M i(s (3 n i (M i (s = min{n i (M i(s, n h i (M i (s}. (4 The ollowing theorem shows that the ambiguity levels o ailure or non-ailure deisions o the deentralized diagnoser given by (1 (4 are upper bounded by N. Theorem 1: Consider the deentralized diagnoser {D i } : L C onsisting o loal diagnosers D i : M i (L C N (i I, and deined by (1 (4. Then, ( s L {D i } φ n(s N. Proo: First, we onsider the ase that {D i } = 1. For any i I suh that n(s = n i (M i (s, i (M i (s = 1. So, n(s = n i (M i(s < n h i (M i (s N + 1,

4 whih implies that n(s N. Next, we onsider the ase that {D i } = 0. For any i I suh that n(s = n i (M i (s, i (M i (s = 0. So, n(s = n h i (M i (s < n i (M i(s N + 1, whih implies that n(s N. We have the ollowing theorem whih states that there are no alse alarms under the deentralized diagnosis perormed using the loal and global diagnosers given by (1 (4. Theorem 2: Consider the deentralized diagnoser {D i } : L C onsisting o loal diagnosers D i : M i (L C N (i I, and deined by (1 (4. Then, ( s L K {D i } (s 0, (5 ( s K {D i } (s 1. (6 Proo: First, we prove (5. Suppose or ontradition that there exists s L K suh that {D i } (s = 0. Then, or any i I suh that n(s = n i (M i (s, i (M i (s = 0. Sine n(s = n i (M i (s = n h i (M i (s < n i (M i(s N + 1, n(s = n h i (M i (s = min{k N M i (s / M i (F k }, whih implies that M i (s / M i (F n(s. It ollows that s / F n(s. Sine s (L K = F 0, there exists l N suh that 0 l < n(s, s F l, and s / F l+1. So, there exists j I suh that s / j M j (H l. It ollows that M j (s / M j (H l. Sine min{k N M j (s / M j (H k } l < n(s < N + 1, n j (M j (s n j (M j(s = min{k N M j (s / M j (H k } < n(s, whih ontradits the deinition o n(s. Next, we prove (6. Suppose or ontradition that there exists s K suh that {D i } (s = 1. Then, or any i I suh that n(s = n i (M i (s, i (M i (s = 1. Sine n(s = n i (M i (s = n i (M i(s < n h i (M i (s N + 1, n(s = n i (M i(s = min{k N M i (s / M i (H k }, whih implies that M i (s / M i (H n(s. It ollows that s / H n(s. Sine s K = H 0, there exists l N suh that 0 l < n(s, s H l, and s / H l+1. So, there exists j I suh that s / j M j (F l. It ollows that M j (s / M j (F l. Sine min{k N M j (s / M j (F k } l < n(s < N + 1, n j (M j (s n h j (M j (s = min{k N M j (s / M j (F k } < n(s, whih ontradits the deinition o n(s. The task o diagnosis urther requires that there are no missed detetions or arbitrarily long ailure traes. I.e., i we let F m := (L K (L KΣ m denote ailure traes in whih a ailure ourred at least m- steps in the past, then it is desired that exists m suh that or all traes in F m, the diagnosis deision is 1, i.e., ( s F m {D i } (s = 1. (7 In the ollowing we establish a neessary and suiient ondition or (7 to hold in terms o the property o N- inerene-diagnosability or ailures. The property o N-inerene-diagnosability or ailures requires that the ailure traes that remain ambiguous ater N levels o inerring must not have inurred a ailure in a ar past (more than a bounded number o steps in the past. Deinition 1: The pair (L, K o languages is said to be N-inerene-diagnosable or ailures i there exists m N suh that F N+1 F m =. We have the ollowing lemma whih states that N- inerene-diagnosability or ailures is a suiient ondition or the nonexistene o missed ailure detetions under the deentralized diagnosis perormed using the loal and global diagnosers given by (1 (4. Lemma 1: Consider the deentralized diagnoser {D i } : L C onsisting o loal diagnosers D i : M i (L C N (i I, and deined by (1 (4. I (L, K is N-inerene-diagnosable or ailures, then {D i } satisies (7 or some m N. Proo: We onsider m N suh that F N+1 F m =. We show by ontradition that {D i } satisies (7 or m. Suppose that there exists s F m suh that {D i } (s 1. Then, there exists i I suh that n(s = n i (M i (s and i (M i (s 1. We have n(s = n h i (M i (s n i (M i(s. (8 Sine s F m and F N+1 F m =, s / F N+1. Also, s F m L K = F 0. There exists l N suh that 0 l N, s F l, and s / F l+1. So, there exists j I suh that s / j M j (H l. It ollows that M j (s / M j (H l. Sine min{k N M j (s / M j (H k } l N < N + 1, n j (M j(s = min{k N M j (s / M j (H k } < N + 1, whih implies that n(s n j (M j(s < N + 1. Sine n(s < N + 1, by (8 that n(s = n h i (M i (s = min{k N M i (s / M i (F k },

5 whih implies that M i (s / M i (F n(s. It ollows that s / F n(s. Sine s F 0, there exists l N suh that 0 l < n(s, s F l, and s / F l +1. So, there exists j I suh that s / j M j (H l. It ollows that M j (s / M j (H l. Sine min{k N M j (s / M j (H k } l < n(s < N + 1, n j (M j (s n j (M j (s = min{k N M j (s / M j (H k } < n(s, whih ontradits the deinition o n(s. The next lemma states that N-inerene-diagnosability or ailures is a neessary ondition or the nonexistene o missed ailure detetions under the deentralized diagnosis perormed using the loal and global diagnosers given by (1 (4. Lemma 2: Consider the deentralized diagnoser {D i } : L C onsisting o loal diagnosers D i : M i (L C N (i I, and deined by (1 (4. I {D i } satisies (7 or some m N, then (L, K is N-inerene-diagnosable or ailures. Proo: We onsider m N suh that {D i } satisies (7. Suppose or ontradition that F N+1 F m. We onsider any s F N+1 F m. By (7, {D i } (s = 1. It ollows rom Theorem 1 that n(s N. Then, s F N+1 F n(s+1. Sine {D i } (s = 1, there exists j I suh that n(s = n j (M j (s and j (M j (s = 1. It ollows that n(s = n j (M j(s N < N + 1. We have n(s = n j (M j(s = min{k N M j (s / M j (H k }, whih implies that M j (s / M j (H n(s. Sine s / j M j (H n(s, s / F n(s+1, whih ontradits the at that s F n(s+1. Armed with Lemmas 1 and 2, we are ready to present the main result o this setion, a theorem that proves the neessity and suiieny o N-inerene-diagnosability or ailures or the nonexistene o missed ailure detetions under the deentralized diagnosis perormed using the loal and global diagnosers given by (1 (4. Theorem 3: Consider the deentralized diagnoser {D i } : L C onsisting o loal diagnosers D i : M i (L C N (i I, and deined by (1 (4. {D i } satisies (7 or some m N i and only i (L, K is N-inerene-diagnosable or ailures. In the ollowing we present a system that is 2- inerene-diagnosable (or ailures but it is not 1-inerenediagnosable. Example 1: We onsider a plant modeled by the inite automaton G shown in Fig. 1(a, whih is a modiied version o the DES onsidered in [13]. Let n = 2, { σ, i σ {a, a M 1 (σ =,, d} ε, otherwise, { σ, i σ {b, b M 2 (σ =,, d} ε, otherwise. Also, let K L be a language generated by the inite automaton R shown in Fig. 1(b. b b a a b b a a d b a (a G d (b R a Fig. 1. Automata G and R o Example 1. We show that (L, K is 2-inerene-diagnosable or ailures. Initially, Sine F 0 = (a + b +d( + a(ε + b + b(ε + a, H 0 = pr( (ab + + ba + + d(a + + b +. M 1 (F 0 = (a + + d( + a + a, M 2 (F 0 = ( + b + d( + b + b, M 1 (H 0 = pr( (a + + a + + d(a + + +, M 2 (H 0 = pr( (b + + b + + d( + + b +, F 1 = F 0 a i M i (H 0 = (a + b + d( + a + b, ( H 1 = H 0 i M i (F 0 = pr( (a + b + d(a + + b +. It ollows that F 1 F m or any m N, whih implies that (L, K is not 0-inerene-diagnosable or ailures. Also, sine M 1 (F 1 = (a + + d( + a + ε, M 2 (F 1 = ( + b + d( + ε + b, M 1 (H 1 = pr( (a + ε + d(a + + +, M 2 (H 1 = pr( (ε + b + d( + + b +, b b

6 F 2 = F 1 i M i (H 1 = (a + b + d( + a + b, ( H 2 = H 1 i M i (F 1 = pr( (a + b + d(a + b. It ollows that F 2 F m or any m N, whih implies that (L, K is not 1-inerene-diagnosable or ailures. Moreover, sine M 1 (F 2 = (a + ε + d( + a + ε, M 2 (F 2 = (ε + b + d( + ε + b, M 1 (H 2 = pr( (a + da, M 2 (H 2 = pr( (b + db, F 3 = F 2 i M i (H 2 = (a + b + d( + a + b, ( H 3 = H 2 i M i (F 2 = H 2. We have F 2 F m = or any m 1, whih implies that (L, K is 2-inerene-diagnosable or ailures. Let N = 2. The loal deisions o D 1 and D 2 omputed using (1 (4 are shown in Table I. For example, D 1 (a is omputed as ollows. By (1 and (2, n 1 (a = 1 and n h 1(a = 2. Sine 1 = n 1 (a < nh 1(a = 2, 1 (a = 1 and n 1 (a = 1, whih implies that D 1 makes a ailure deision ollowing the observation a M 1 (L with the ambiguity level 1. TABLE I LOCAL DECISIONS OF D 1 AND D 2. t M 1 (L n 1 (t nh 1 (t 1(t n 1 (t t 3 3 φ 3 t a 3 3 φ 3 t a t a t = d 3 3 φ 3 t d t da 3 3 φ 3 t da t da t M 2 (L n 2 (t nh 2 (t 2(t n 2 (t t 3 3 φ 3 t b 3 3 φ 3 t b t b t = d 3 3 φ 3 t d t db 3 3 φ 3 t db t db Then, the global diagnosis deisions o the deentralized diagnoser {D i } are omputed as shown in Table II. For example, {D i } (a is omputed as ollows. Sine 1 = n 1 (M 1 (a < n 2 (M 2 (a = 3 and 1 (M 1 (a = 1, n(a = 1 and {D i } (a = 1. By Table II, we an veriy that {D i } satisies (5, (6, and (7 or m 1. TABLE II GLOBAL DECISIONS OF {D i }. s L n(s {D i } (s s 3 φ s a(ε + 3 φ s b(ε + 3 φ s a s b s ab 0 0 s ba 0 0 s d(ε + 3 φ s d s da(ε + 3 φ s db(ε + 3 φ s da s db s dab 0 1 s dba 0 1 Remark 1: In the system o Example 1, the event represents the ailure event. By the examples shown in [13], this system is not onditionally odiagnosable or the ailure. However as we showed above the system is 2-inerenediagnosable or the ailure. V. COMPUTATION/VERIFICATION OF INFERENCE-BASED DECENTRALIZED DIAGNOSIS/DIAGNOSERS The omputation o loal deisions using (1 (4 requires omputing the sequene o language pairs {(F k, H k }, and we present a reursive method or omputing it. Let G = (X, Σ, α, X 0, X be the plant model with L(G = L m (G = L, and R = (Y, Σ, β, Y 0, Y be a deterministi generator o the non-ailure speiiation language, i.e., L(R = L m (R = K = H 0 An aeptor or F 0 = L K is the automaton G R, where R = (Y {F }, Σ, β, Y 0, Y is R ompleted by (i adding a dump state F, and (ii adding a transition on eah event at a state in Y {F } to the dump state F i that event is not deined at that state in R. Then it an be veriied that traes in G R that end at a state with seond oordinate F belong to L K. Also, L(G R = L(G L( R = L(G Σ = L(G. Let R Fk and R Hk denote the inite aeptors o F k and H k, respetively. For eah i I, a inite aeptor o i M i (F k is onstruted as ollows: Repliate eah transition that exists in R Fk by a set o transitions on all M i -indistinguishable events. Note that sine an ε-transition is impliitly deined at eah state as a sel-loop, unobservable events will get added as sel-loops at eah state o R Fk. Then, the resulting, possibly nondeterministi, automaton M i (F k. It should be noted that this resulting automaton, denoted by i M i (R Fk, has the same state set as R Fk. In the same way, we an onstrut a inite automaton aepting i M i (H k, denoted by i M i (R Hk. Then, aepts i

7 the synhronous ompositions R Fk ( i M i (R Hk and R Hk ( i M i (R Fk aept F k+1 and H k+1, respetively. Let Y Fk and Y Hk be the state sets o R Fk and R Hk, respetively. The languages F k+1 and H k+1 are omputed rom F k and H k in O( Y Fk Y Hk I and O( Y Hk Y Fk I, respetively. The veriiation o N-inerene-diagnosability or ailures requires the heking the existene o m suh that F N+1 F m = F N+1 (L KΣ m =. For this we onsider the automaton R FN+1 G R, where R FN+1 = (Y FN+1, Σ, β FN+1, Y 0,FN+1, Y m,fn+1 is an aeptor o F N+1 obtained as outlined above, G = (X, Σ, α, X 0, X is the plant model with L(G = L m (G = L, R = (Y, Σ, β, Y 0, Y is a deterministi generator o the non-ailure speiiation language, i.e., L(R = L m (R = K, and R = (Y {F }, Σ, β, Y 0, Y is the ompletion o R by way o inlusion o an additional state F and ertain additional transitions as outlined above. Note that L(R FN+1 G R = L(R FN+1 L(G R = pr(f N+1 L(G = pr(f N+1. To veriy N-inerene-diagnosability or ailures, we veriy in the automaton R FN+1 G R whether (i the region o attration o states with the irst oordinated marked in R FN+1 (these are states reahed by traes in F N+1 ontains all states with the third oordinate F (these are states reahed by traes in L K, and (ii all strongly onneted omponents o R FN+1 G R that ontain a state with the irst oordinate marked in R FN+1 are singletons and with no sel-loops. This holds i and only i all traes in F N+1 have the property that they are inite extensions o boundary ailure traes, (L K := (L K KΣ, i.e., F N+1 (L K Σ p or some p 0, whih an be seen to be equivalent to the property o N-inerene diagnosability or ailures. Sine the above mentioned region o attration as well as the set o srongly onneted omponents an be omputed linearly in the size o R FN+1 G R [2], the omplexity o heking N-inerene-diagnosability is O( Y FN+1 X Y. VI. PROPERTIES OF N -INFERENCE-DIAGNOSABILITY In this setion, we study various properties o N-inerene diagnosable systems or ailures. First, we show that the lass o odiagnosable systems studied in [5] is equivalent to the lass o 0-inerene-diagnosable systems or ailures. Deinition 2: [5] The pair (L, K o languages is said to be odiagnosable i ( m N ( s L K( st L K s.t. t m ( i I( u E i (st; u L K, where E i (st := i M i (st L. Theorem 4: The pair (L, K o languages is 0-inerenediagnosable or ailures i and only i it is odiagnosable. Proo: ( We assume that (L, K is 0-inerenediagnosable or ailures. Then, there exists m N suh that F 1 F m =. Let s L K. Consider any st L K suh that t m. We have st (L K (L KΣ m = F m. Sine F 1 F m = and st F m F 0, st F 0 F 1, whih implies that there exists i I suh that st / i M i (H 0 = i M i (K. It ollows that M i (st / M i (K. For any u E i (st, M i (u = M i (st / M i (K, whih implies that u / K, i.e., u L K. Thus, (L, K is odiagnosable. ( We assume that (L, K is odiagnosable. Let m N be a nonnegative integer suh that the ondition o odiagnosability holds. Suppose or ontradition that F 1 F m. We onsider any s F 1 F m. Sine s F m, we an write s := tu L K where t L K and u m. Also, sine s F 1, tu i M i (K or all i I. There exists v i K L suh that M i (tu = M i (v i. Sine v i E i (tu and v i K or all i I, the ondition o odiagnosability does not hold. This is a ontradition. Next, we show that the notion o onditional F- odiagnosability introdued in [13] is similar to that o 1- inerene-diagnosability or ailures. Deinition 3: [13] The pair (L, K o languages is said to be onditionally F-odiagnosable i where ( m N ( s L K( st L K s.t. t m ENF j (u ( i I( u E i (st (u K [( j I( v ENF j (u; v K], := {s E j (u the last event o s is not in Σ i,uo }. Theorem 5: The pair (L, K o languages is 1-inerenediagnosable or ailures i and only i ( m N ( s L K( st L K s.t. t m ( i I( u E i (st (u K [( j I( v E j (u; v K]. Proo: ( We assume that (L, K is 1-inerenediagnosable or ailures. Then, there exists m N suh that F 2 F m =. Let s L K. Consider any st L K suh that t m. We have st (L K (L KΣ m = F m. Suppose or ontradition that ( i I( u i E i (st K ( j I( v ij E j (u i ; v ij L K. Sine st F 0 and st i i M i (H 0 or all i I, st F 1. Also, sine u i H 0 and u i M i (u i j M j (v ij j M j (F 0 or all j I, u i H 1. It ollows that st i M i (u i i M i (H 1 or all i I, whih implies together with st F 1 that st F 2. Thus, st F 2 F m, whih ontradits the assumption that (L, K is 1-inerene-diagnosable or ailures. ( Let m N be a nonnegative integer suh that the ondition o Theorem 5 holds. Suppose or ontradition that F 2 F m. We onsider any s F 2 F m. Sine s F m, we an write s := tu L K where t L K and u m. Also, sine s F 2, tu i M i (H 1 or all i I. There exists v i H 1 suh that M i (tu = M i (v i. It ollows that v i E i (tu and v i K or all i I. Moreover, sine

8 v i H 1, v i j M j (F 0 or all j I. There exists w ij F 0 suh that M j (v i = M j (w ij. It ollows that w ij E j (v i and w ij L K or all j I. This ontradits the ondition o Theorem 5. Sine ENF i (s E i (s or any s L and any i I, the ollowing is an immediate orollary o Theorem 5. Corollary 1: I the pair (L, K o languages is 1- inerene-diagnosable or ailures, then it is onditionally F- odiagnosable. Remark 2: Theorem 5 shows that i ENF j (u is replaed by E j (u in the deinition o onditional F-odiagnosability, then the notions o 1-inerene-diagnosability or ailures and onditional F-odiagnosability are idential. We also establish that the lasses o N-inerenediagnosable systems or ailures orm a monotonially inreasing sequene as a untion o N. Sine the sequene o language pairs {(F k, H k } is monotonially dereasing or any k N, the ollowing result is easily obtained (the proo is omitted. Theorem 6: For any N N, i the pair (L, K o languages is N-inerene-diagnosable or ailures, then it is (N + 1-inerene-diagnosable or ailures. The onverse relation o Theorem 6 need not hold. For example, the system o Example 1 is 2-inerene-diagnosable or ailures, but not 1-inerene-diagnosable. VII. CONCLUSION A key issue in deentralized deision-making is the usion o the loal deisions to arrive at a global deision. A key observation we made in a prior work [4] is that suh deision usion an be ailitated by assessing ambiguity levels o eah loal deision-maker (arising due to its limited sensing apability and using that knowledge to arrive at a global deision. Our prior work [4] used this idea to propose an inerene-based ramework or deentralized ontrol (inerening over sel-ambiguities and those o others is used or the assessment o the ambiguity level o ones own deision. The present paper proposes an inerene-based ramework or deentralized diagnosis. The proposed ramework is able to extend the prior approahes to deentralized diagnosis (suh as those reported in [5], [13]. Also through the work reported in [6] it is evident that a deentralized diagnosis ramework (one involving no ommuniation among loal diagnosers is also useul or distributed diagnosis appliations (one involving ommuniation among loal diagnosers, and so the appliability o the proposed ramework extends to the setting o distributed diagnosis. It should be noted that as the order o inerening inorporated into deentralized deision-making is enhaned, the orresponding ost o omputing the loal deisions is also inreased (as ormalized in Setion V. So the additional gain resulting rom a higher order o inerening omes at an additional omputational ost. REFERENCES [1] R. K. Boel and J. H. van Shuppen. Deentralized ailure diagnosis or disrete-event systems with onstrained ommuniation between diagnosers. In Proeedings o International Workshop on Disrete Event Systems, [2] Y. 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