Verification of Coprognosability in Decentralized Fault Prognosis of Labeled Petri Nets

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1 28 IEEE Conference on Decision and Conrol (CDC Miami Beach, FL, USA, Dec. 7-9, 28 Verificaion of Corognosabiliy in Decenralized Faul Prognosis of Labeled Peri es Wenqing Wu, Xiang Yin and Shaoyuan Li Absrac We invesigae he roblem of decenralized faul rognosis of discree-even sysems modeled by unbounded labeled Peri nes. We assume ha he sysem is moniored by a se of local agens (rognosers wih local observaions so ha hey can redic he occurrence of faul in he sysem as a eam. I is known in he lieraure ha he noion of corognosabiliy rovides he necessary and sufficien condiion for he exisence of a se of decenralized rognosers so ha any faul can be rediced before is occurrence wihou false alarm. In his aer, we invesigae he verificaion of corognosabiliy for sysems modeled by labeled Peri nes. We show ha corognosabiliy is decidable even when he Peri ne is unbounded. Secifically, we rovide an aroach o ransform he corognosabiliy verificaion roblem o a model checking roblem ha can be effecively solved. Our resul exends exising works on corognosabiliy analysis in decenralized faul rognosis from regular languages o Peri ne languages. I. ITRODUCTIO We invesigae he faul rognosis roblem for Discree- Even Sysems (DES modeled by labeled Peri nes. This roblem is crucial in many safey-criical sysems, where we need o redic he occurrences of fauls so ha some roecing acions can be aken before a faul is encounered. In his aer, we consider DES modeled by labeled Peri nes, a comuaional model ha is widely used in modeling cyber-hysical sysems wih concurrency, e.g., flexible manufacuring sysems and sofware sysems. Our goal is o analyze a riori wheher or no any faul in he sysem can be successfully rediced. In he conex of DES, he roblem of faul rognosis has drawn considerable aenion in he as years; see, e.g., [6], [], [5], [6], [9], [2], [23] [25], [28], [3] [32]. This roblem was iniially sudied in [2], [3], where he auhors consider DES modeled by finie-sae auomaa and a language-based roery called redicabiliy (or rognosabiliy was roosed. Secifically, redicabiliy is he necessary and sufficien condiion under which here exiss a cenralized rognoser ha can always issue a faul alarm before he occurrence of faul wihou any false alarm. Recenly, redicabiliy/rognosabiliy analysis has also been invesigaed for DES modeled by imed auomaa [9], sochasic auomaa [6], [], [], [9] and fuzzy auomaa [5]. For DES modeled by Peri nes, he auhors in [] [3], This work is suored by he aional aural Science Foundaion of China (683259, W. Wu, X. Yin and S. Li are wih Dearmen of Auomaion and Key Laboraory of Sysem Conrol and Informaion Processing, Shanghai Jiao Tong Universiy, Shanghai 224, China. {wuwenqing.dy,yinxiang,syli}@sju.edu.cn. [7] have invesigaed online faul rognosis mehods for boh logical and sochasic Peri nes. Recenly, i has been shown in [27] he condiion of rognosabiliy is decidable for unbounded Peri nes in he cenralized seing. In many large-scale sysems, he informaion srucure are naurally decenralized. For such sysems, cenralized faul rognosis algorihms canno be direcly alied due o he informaion consrain and we need o develo corresonding decenralized rognosic roocols for he urose of faul rognosis. Therefore, he roblem of decenralized faul rognosis has also drawn many aenion in he as years; see, e.g., [4] [6], [22], [3], [32]. In his seing, i is assumed ha he sysem is moniored by a se of local rognosers. Each local rognoser has is own observaion and can send local rognosic decision o a coordinaor, where a final global rognosic decision is issued. Paricularly, in [6], he auhors invesigaed he decenralized faul rognosis roblem under he disjuncive archiecure, where a global faul alarm is issued if one local faul alarm is issued. Moreover, he auhors in [6] roosed a noion called corognosabiliy as he necessary and sufficien condiion under which here exiss a decenralized rognoser ha can successfully redic faul in he disjuncive archiecure. In [5] and [22], he decenralized faul rognosis roblem has been furher sudied in he conjuncive archiecure and he inference-based archiecure, resecively. In his aer, we sudy he decenralized faul rognosis roblem for (unbounded labeled Peri nes in he disjuncive archiecure. Secifically, we are ineresed in verifying corognosabiliy for Peri ne languages in order o deermine a riori if he faul rognosic ask can be accomlished. oe ha, since he sae sace of a Peri ne is unbounded in general, exising aroaches based on finie-sae auomaa canno be alied o our roblem. The main conribuions of his aer are as follows. Firs, we rovide a necessary and sufficien condiion for corognosabiliy of DES modeled by unbounded Peri nes. Then we resen an effecive consrucion o caure his condiion and show ha checking corognosabiliy is decidable for general unbounded labeled Peri nes. To he bes of our knowledge, exising works on decenralized faul rognosis menioned above assume ha he sysems are modeled by finie-sae auomaa and he decenralized faul rognosis roblem has never been sudied for sysems modeled by Peri nes. Therefore, our resuls exend exising works on corognosabiliy analysis from regular languages o Peri nes languages. The remaining ar of he aer is organized as follows. Secion II resens necessary reliminaries. In Secion III, /8/$3. 28 IEEE 4845

2 corognosabiliy is defined for labeled Peri nes and is relevan roeries are also discussed. In Secion IV, we rovide a necessary and sufficien condiion for corognosabiliy, which can be effecively checked by an exising model checking roblem. Finally, we conclude he aer in Secion V. II. PRELIMIARIES A (lace/ransiion ne is a direced biarie grah, defined as a 4-ule = (P, T, A, ω, where P = {, 2,..., n } is he finie se of laces, T = {, 2,..., m } is he finie se of ransiions, A (P T (T P is he se of direced arcs from laces o ransiions and from ransiions o laces and ω : A is he weigh funcion ha assigns o each arc a non-negaive ineger. We inroduce he emy ransiion λ and we define T + := T {λ}. For any lace P, we denoe is rese by I(, i.e., I( := { T : (, A}; we denoe is osse by O(, i.e., O( := { T : (, A}. For any ransiion, we define is rese I( and is osse O( analogously, which are ses of laces. Given a ne, a marking M is a vecor M = [M( M( 2... M( n ] n, where M( i is he number of okens in lace i P. A Peri ne is a air, M, where is a ne and M n is he iniial markings. We say ha ransiion is enabled a sae M if I( : M( ω(,. If is enabled a M, hen i can fire and yields a new marking M defined by P : M ( = M( ω(, + ω(,. We denoe by M ha ransiion T is enabled a M in ne and denoe by M M ha he firing of a M yields M in ne. We will also omi he subscri when he ne is clear from he conex. Le T denoe he se of all finie sequences of ransiions (or, for simliciy, sequences including he emy ransiion λ. For any T, we have λ = λ =. A sequence = 2... k T is said o be enabled a M if i {,..., k} : M i i, where M = M and M i i Mi+. Analogously, we denoe by M ha T can be fired a M and by M M ha firing yields M. For a Peri ne, M, L(, M denoes he se of finie sequences which are enabled a M, i.e., L(, M = { T : M }. Given a sequence T, we denoe by he se of refixes of, i.e., = { i T : j T s.. i j = }. Finally, he lengh of sequence is denoed by. For any sequence T and any ransiion T, we denoe by # ( he number of imes occurs in. We define Σ as a finie se of evens (or alhabes. A sring is a finie sequence of evens and we use Σ o denoe he se of all srings including he emy sring ɛ. A labeled Peri ne is a rile, M, L, where, M is a Peri ne and L : T Σ {ɛ} is a labeling funcion ha assigns o each ransiion an even. In oher words, for any T, L( indicaes he even ha can be observed when fires. Given a ransiion T, we call i observable if L( Σ; oherwise, is unobservable. Therefore, T is naurally ariioned as T = T o T uo, where T o and T uo are he ses of observable and unobservable ransiions, resecively. The labeling funcion L can also be exended from T o T recursively by: (i L(λ = ɛ; and (ii T, T : L( = L(L(. Therefore, he language generaed by labeled Peri ne, M, L is a se of srings defined by L(L(, M := {L( : L(, M }. Finally, we make he following sandard assumion in he analysis of arially-observed DES A, M does no ener a deadlock, i.e., ( T : M M( T [M ]. III. COPROGOSABILITY OF LABELED PETRI ETS In he decenralized faul rognosis roblem, he sysem, M is moniored by a se of n local agens (or local rognosers. We denoe by I = {, 2,..., n} he index se. We assume ha each agen has is own observaion. Formally, for any i I, we denoe by L i : T Σ {ɛ} is local labeling funcion and we denoe by T o,i and T uo,i he ses of observable ransiions and unobservable ransiions w.r.. L i, resecively. Therefore, a labeled Peri ne in he decenralized seing is wrien by, M, {L i } i I. In he faul rognosis roblem, we assume ha he se of ransiions is furher ariioned as wo disjoin ses T = T T F, where T is he se of normal ransiions and T F is he se of faul ransiions. For any sequence = 2... k T, wih a sligh abuse of noaion, we denoe by T F if conains a faul ransiion, i.e., i {,..., k} : i T F. In order o characerize wheher or no any faul in a decenralized sysem can be rediced, a language-based condiion called corognosabiliy was roosed in [6]. This definiion is reviewed as follows in he conex of Peri ne languages. Definiion 3. (Corognosabiliy: Le, M, {L i } i I be a labeled Peri ne wih decenralized informaion. Then, M, {L i } is said o be corognosable w.r.. T F if ( α L(, M : T F α( β α : T F / β ( i I( θ L(, M : L i (θ = L i (β T F / θ ( K ( θγ L(, M [ γ K T F γ] ( Inuiively, corognosabiliy requires ha for any faul sequence, i mus have a non-faul refix such ha, a leas one local rognoser knows for sure ha a faul will occur in a finie number of ses. Therefore, a local faul alarm can be sen o he coordinaor in order o issue a global faul alarm before he faul acually occurs. I has been shown in [6] ha corognosabiliy rovides he necessary and sufficien condiion under which here exiss a se of local rognosers ha can achieve he following wo requiremens under he disjuncive archiecure Any faul can be rediced before is occurrence; and A faul will occur in a finie number of ses once a faul alarm is issued. In he disjuncive archiecure, he coordinaor issues a global alarm if one local rognoser issues a faul alarm. 4846

3 2 2 K[M ]. 3 ε, ε 3 ε, ε Inuiively, a boundary marking is a marking from which a faul can occur in he nex se and a non-indicaor marking is 2 f b, b 2 f a marking from which an arbirarily long non-faul sequence (a Peri ne, M, {L, L 2 } can occur. Since a marking is a vecor wih is elemens aken from he se of nonnegaive inegers, here does no exis a 2 2 sricly monoone decreasing sequence of markings. Hence, ε, ε 3 under ε, ε he deadlock-free assumion, M is a non-indicaor b, b, ε 3 3 ε, ε marking if and only if b,2 2 f b, b 2 f 2 f, 2 T s.. M M 2 M2 and M M 2. a, (b Peri ne, M, {L 2, Lf 2 } a,2 ε, a ex, we inroduce he following lemma ha illusraes Fig.. For each ransiion, is associaed label (e, e 2 means ha how o decide corognosabiliy according o he above wo L ( = e and L 3 ( = e noions. b, b, ε 3 3 Lemma ε, ε 3.: Labeled Peri ne, M b,2, {L i } i I is no corognosable w.r.. T F, if and only if, here exis n + The reader is referred o [6] for how corognosabiliy 2 f non-faul sequences B a,,,..., n T such ha guaranees he above wo condiions. In his aer, we will 2 f M focus on how o verify his condiion for languages generaed B is a a,2 boundary ε, a marking, where M B MB ; and 2 For any i I, M by unbounded Peri nes. i is a non-indicaor marking, where M Firs, le us illusrae he noion of corognosabiliy in Peri i Mi ; and 3 For any i I, L nes by he following examles. i ( B = L i ( i. Examle 3.: Le us consider labeled Peri ne IV. VERIFICATIO OF COPROGOSABILITY, M, {L, L 2} wih wo local agens shown in Figure (a, where T o, = {, 3 }, T o,2 = {, 2, 3 } and T F = {f }. Le Σ = {}, L ( = a, L ( 3 = b and L 2( = L 2( 2 = L 2( 3 = b. oe ha ransiion 3 has o fire before he occurrence of faul ransiion f, and once i fires, he firs agen observes even L ( 3 = b, which can only be generaed by ransiion 3 hrough labeling funcion L, i can issue a faul alarm unambiguously. Hence, he sysem is corognosable, alhough from he ersecive of L 2, all he observable ransiions have he same label. Examle 3.2: Le us consider labeled Peri ne, M, {L, L 2 } wih wo local agens shown in Figure (b, where T o, = {, 3 }, T o,2 = { 2, 3 } and T F = {f }. Le Σ = {}, L ( = L ( 3 = a and L 2 ( 2 = L 2 ( 3 = b. This sysem is no corognosable. To reveal his, we consider faul sequence 3 f L(, M. Then for 3 3 f : T F / 3, we can find L(, M such ha L ( = L ( 3 = a and an arbirarily long ( non-faul sequence M k, k is defined. Similarly, for 3, we can also find 2 L(, M such ha L 2 ( 2 = L 2 ( 3 = b and an arbirarily long non-faul 2( 2 sequence M k, k is defined. In oher words, when 3 occurs, none of he agens knows for sure ha a faul will occur in a finie number of ses. Therefore, faul sequence 3 f canno be alarmed correcly, i.e., he sysem is no corognosable. Before roceeding o he verificaion of corognosabiliy for labeled Peri nes, we firs define he noions of boundary marking and non-indicaor marking, which was originally inroduced in [27]. Definiion 3.2: A marking M n is said o be a boundary marking if ( f T F [M f ]; and a non-indicaor marking if ( K ( T : In he revious secion, we have discussed he definiion of corognosabiliy for labeled Peri nes and rovided an aroach o characerize corognosabiliy based on he noions of boundary markings and non-indicaor markings. In his secion, we rovide a verifiable necessary and sufficien condiion for corognosabiliy in order o show ha corognosabiliy is indeed decidable. According o Lemma 3., he verificaion of corognosabiliy is equivalen o checking he exisence of a leading sequence B ha goes o a boundary marking and n following sequences,..., n, each of hem goes o a nonindicaor marking, such ha B and i look he same under he i-h observaion maing L i. To his end, we need o consruc a new ne ha racks all such leading and following sequences. Le, M, {L i } i I be he original labeled Peri ne. Firs, we define a new labeled Peri ne as follows: Ñ, M, { L i } i I, where Ñ = ( P, T, Ã, ω P = P {f }, where f is a new lace; T = T { e,i : e Σ, i I}, where each e,i is a new ransiion defined for each e Σ, i I; Ã and ω are defined by: For any T, I( and O( are he same as in and P : ω(, = ω(,, ω(, = ω(,. For any T F, I( is he same as in wih P : ω(, = ω(,, while O( = { f } wih ω(, f =. For any e,i, e Σ, i I, I( = O( = { f } and ω( e,i, f = ω( f, e,i =. The iniial marking is M = [M ], where we assume ha he las lace is f. 4847

4 2 f b, b 2 f ε Fig b, b, ε 2 3 ε, ε ε, ε a, ε a, 2 b f 2 b,2 f f a,2 (a Ñ, M, { L, L 2 } (b, M, {L, L 2 } ε, a a, 2 3 b, b 2 and Ñ are consruced based on he sysem in Figure (b. For each i I, he labeling funcion L i : T Σ {ɛ} is defined as L i ( if T ɛ if T f L i ( = e if = e,j and i = j ɛ if = e,j and i j Inuiively, Ñ is a coy of exce for new lace f and new ransiions e,i. In aricular, for any non-faul sequence, he dynamic of Ñ and he labels coincide wih hose of Ñ. On he oher hand, when a faul ransiion fires, a oken is sen o new lace f in Ñ signifying he occurrence of faul. For each e Σ, a series of self-loo ransiions e,i are corresondingly defined a f, so ha for any non-faul sequence whose label is e... e k under L i, here always exiss a self-loo sequence in he form of e,i... ek,i ha roduces he same observaion under L i bu roduces emy observaion under L j, j i. As will become clear laer, such a consrucion is used o find non-indicaor markings. Also, we denoe by he normal ne obained by removing ransiions in T F from. We denoe by,,...,,n n coies of wih disjoin laces P,i and ransiions T,i. For examle, for Peri ne shown in Figure (b, Ñ and are shown in Figures 2(a and 2(b, resecively. ex, we build a new (unlabeled Peri ne, M, which synchronizes Ñ, M wih,, M,...,,n, M so ha he move in he firs comonen and he (i + -h comonen have he same label under L i and L i. Formally, Peri ne, M,, where = (P, T, A, ω, is defined as follows: P = P P, P,2 P,n ; T T + T +, T +,2 T +,n \ {(λ,... }; (n+imes A and ω are defined by: For any T, ransiion (,, 2,..., n T ε, ε f is defined if, for any i I, we have [ L i ( = L i ( i ] [ L i ( =ɛ i =λ] (2 For each i I, ransiion (,,..., i,..., n T is defined if [L i ( i =ɛ] [ i λ] [ j, j i : j =λ] (3 b, b, ε 3 ε, ε For each defined ransiion b,2 = (,, 2,..., n T, we have I( = f i a, 2 f I( i and O( = i O( i. Also, a,2 ε, a { ω(, if ω (, = P ω (, = ω,i ( i, { ω(, ω,i (, i M, = [ M M M... M ]. if P,i if P if P,i Inuiively, ne, M, consiss of (n+ comonens, where he firs comonen Ñ is used o rack he leading sequence ha goes o a boundary marking and he (i + - h comonen,i is used o rack he i-h following sequence ha goes o a non-indicaor marking. Moreover, for each i I, he leading sequence and he i-h following sequence have he same observaion based on he labeling funcions L i and L i. More secifically, for any ransiion = (,,..., i,..., n T saisfying Equaion (2, he leading ransiion moves and each following comonen should move accordingly if his ransiion does no look as ɛ locally. Similarly, for any ransiion = (,,..., i,..., n T saisfying Equaion (3, each following comonen can execue a locally unobservable ransiion wihou involving he leading comonen and oher following comonens. To sum u, for any sequence = (,, 2,..., n L(, M, we can conclude ha Li ( = L i ( i. On he oher hand, for any L(Ñ, M, i L(,i, M and L i ( = L i ( i, i I, here exiss a sequence L(, M, such ha i is in he form of (,, 2,..., n (ossibly by insering λ. This consrucion of is moivaed by he so called verifier (or win-machine consrucion for diagnosabiliy and rognosabiliy analysis; see, e.g., [7], [8], [8], [2], [29]. Here, we ado he basic idea and exend i o he decenralized seing for he urose of faul rognosis wih non-rivial modificaions. The following heorem reveals how o use ne, M, o es corognosabiliy for labeled Peri nes. Theorem 4.: Labeled Peri ne, M, {L i } i I is no corognosable w.r.. T F, if and only if, here exiss a sequence M, α M β M 2 (4 4848

5 in, M, such ha (M 2 M ( T F {λ} {λ} # α ( (5 ( # β ( i I T + T +, T +,i T,i T +,i+ T +,n Le us exlain he basic idea of he above heorem. oe ha, each sequence in consiss of (n + comonens. We denoe by α i he i-h comonens in α, i.e., α = (α, α,..., α n ; he same for β. ow, le us assume ha here exiss a sequence saisfying he hree condiions. Since T F {λ} {λ} # α(, we know ha he firs comonen α mus conain a faul ransiion and here exiss a refix of α leading o a boundary marking. Also, he las condiion imlies ha β, β 2,..., β n are all non-λ. This ogeher wih M 2 M guaranees ha any marking reached by a refix of α, α 2,..., α n is a non-indicaor marking, since each β i can be fired for an arbirary number of imes. Thus, he condiions in Theorem 4. essenially ensure ha here exis a leading sequence ha goes o a boundary marking and n following sequences, each of hem goes o a non-indicaor marking, such ha he leading sequence and each following sequence are locally observaionally equivalen, which disroves corognosabiliy. On he oher hand, if he labeled Peri ne is no corognosable, i.e., α here exiss a sequence α such ha M, M = [MB M, M,2... M,n], where M B is a boundary marking and M,, M,2,..., M,n are n non-indicaor markings. As all faul ransiions are unobservable in L, we have ( f,λ,...,λ M M 2 for some f T F. Such an M 2 can be exended o a covering since he self-loos in he form of e,i in Ñ can rack any sequence ha forms a covering for each comonen. In he above heorem, we have shown ha checking noncorognosabiliy is equivalen o checking he exisence of a secific sequence in ne, M,. This condiion is acually verifiable as a secial case of Peri nes model checking roblem called he Yen s roblem [26]. This leads o he decidabiliy of corognosabiliy. Theorem 4.2: Checking corognosabiliy is decidable for label Peri nes. Moreover, i is in EXPSPACE. Proof: The decidabiliy comes from he fac ha he exisence of a sequence saisfying he Equaion (5 is exressed by a logic called Yen s formula [4], [26], whose saisfiabiliy is decidable. Secifically, Yen s resul allows us o es he exisence of a sequence M 2 M k Mk M k such ha a redicae φ(m,..., M k,,..., k holds, where φ(m,..., M k,,..., k is a marking and ransiion redicae obained by conjuncions and disjuncions of he erms in he form of M i ( = M j (, M i ( > M j (, # i ( c, # i ( c and # i ( # j (, where, T,, P and c is an arbirary consan. For more deail on Yen s logic, he reader is referred o he lieraure [26]. Moreover, i has been shown in [4] ha he Yen s roblem can be decided in EXPSPACE if M M k and here is no ransiion redicae. Clearly, our condiion in Theorem 4. saisfies he Yen s formula and M M 2. oe ha ransiions redicaes can be relaced by marking redicaes. Moreover, he size of our condiion is olynomial in he number of ransiions and number of laces in he original ne (bu is exonenial in he number of local agens. Therefore, we conclude ha checking corognosabiliy is decidable and i can be solved in EXPSPACE in he size of he original ne when he number of local agens is fixed. Finally, we show how o use Theorem 4. o verify corognosabiliy by he following examle. Examle 4.: Le us again consider labeled Peri ne, M, {L, L 2 } shown in Figure (b, where T o, = {, 3 }, T o,2 = { 2, 3 } and T F = {f }. Is corresonding ne, M, is deiced in Figure 3. For he sake of clariy, we use suer-scri i for each ransiion and lace ha corresond o Ñ,i. For examle, ransiion is defined (, in defined in, since L ( = a and L 2 ( = ɛ, i.e., Equaion (2 is fulfilled. Similarly, ransiion (λ, λ, 2 is also defined since L 2( = ɛ; his is he siuaion described in Equaion (3. As we discussed in Examle 3.2, he sysem is no corognosable. ex we show his using Theorem 4.. Secifically, we have he following sequence in, M, : =:M, =:α { }} { ( 3,, 2 2 (f, λ =:M =:β { }} { ( a,, =:β 2 { }} { ( b,2, λ, 2 2 =:M 2 where laces in each marking are ordered by {, 2, f,, 2, 2, 2 2 }. This sequence saisfies he condiion in Equaion (5. Firs, we have (f, λ T F {λ} {λ} and # α ((f, λ =. Second, for β := β β 2, we have ( a,, T + T, T +,2, ( b,2, λ, 2 2 T + T +, T,2 and # β (( a,, = # β (( b,2, λ, 2 2 =. Also, i is clear ha M 2 = M. Therefore, he sysem is no corognosable according o Theorem 4.. V. COCLUSIO In his aer, we solve he roblem of verifying corognosabiliy for decenralized labeled Peri nes. We show ha his roblem is decidable by roviding a verifiable necessary and sufficien condiion. Our resul exends exising resuls on corognosabiliy analysis from regular languages o Peri ne languages. oe ha corognosabiliy is a condiion ha deermines a riori wheher or no any faul can be rediced. How o develo efficien online rognosic algorihms for decenralized Peri nes is an imoran fuure direcion. REFERECES [] R. Ammour, E. Leclercq, E. Sanlaville, and D. Lefebvre. Faul rognosis of imed sochasic discree even sysems wih bounded esimaion error. Auomaica, 82:35 4,

6 ( 3,, 3 2 ε(a (λ, 2 2 (, (, 3 ( 3, 3, 3 2 ( 3,, 2 ( a,, ( a,, 3 ( 2, λ, 2 2 ( 2, λ, (f, λ f ( b,2, λ, (λ, λ, 2 2 ( 3, 3, 2 2 ( b,2, λ, 2 2 Fig. 3., M, for he sysem in Figure (b. [2] R. Ammour, E. Leclercq, E. Sanlaville, and D. Lefebvre. Sae esimaion of discree even sysems for rul redicion issue. Inernaional Journal of Producion Research, 55(23:74 757, 27. [3] R. Ammour, E. Leclercq, E. Sanlaville, and D. Lefebvre. Fauls rognosis using arially observed sochasic eri-nes: an incremenal aroach. Discree Even Dynamic Sysems, ages 2, 28. [4] M. F. Aig and P. Habermehl. On Yen s ah logic for Peri nes. Inernaional Journal of Foundaions of Comuer Science, 22(4: , 2. [5] B. Benmessahel, M. Touahria, and F. ouioua. Predicabiliy of fuzzy discree even sysems. Discree Even Dynamic Sysems, 27(4:64 673, 27. [6]. Berrand, S. Haddad, and E. Lefaucheux. Foundaion of diagnosis and redicabiliy in robabilisic sysems. In 34h IARCS Annual Conf. FSTTCS, ages , 24. [7] B. Bordbar, A. Al-Ajeli, and M. Alodib. On diagnosis of violaions of consrains in Peri ne models of discree even sysems. In 26h Inernaional Conference on Tools wih Arificial Inelligence, ages IEEE, 24. [8] M.P. Cabasino, A. Giua, S. Laforune, and C. Seazu. A new aroach for diagnosabiliy analysis of Peri nes using verifier nes. IEEE Trans. Auomaic Conrol, 57(2:34 37, 22. [9] F. Cassez and A. Grasien. Predicabiliy of even occurrences in imed sysems. In Formal Mod. Anal. Timed Sys., ages [] M. Chang, W. Dong, Y. Ji, and L. Tong. On faul redicabiliy in sochasic discree even sysems. Asian journal of Conrol, 5(5: , 23. [] J. Chen and R. Kumar. Sochasic failure rognosabiliy of discree even sysems. IEEE Trans. Auom. Conr., 6(6:57 58, 25. [2] S. Genc and S. Laforune. Predicabiliy of even occurrences in arially-observed discree-even sysems. Auomaica, 45(2:3 3, 29. [3] T. Jéron, H. Marchand, S. Genc, and S. Laforune. Predicabiliy of sequence aerns in discree even sysems. In Proc. 7h IFAC World Congress, ages , 28. [4] A. Khoumsi. Muli-decision rognosis: Decenralized archiecures cooeraing for redicing failures in discree even sysems. In 2h Medierranean Conference on Conrol & Auomaion, ages IEEE, 22. [5] A. Khoumsi and H. Chakib. Conjuncive and disjuncive archiecures for decenralized rognosis of failures in discree-even sysems. IEEE Trans. Auom. Sci. Engin., 9(2:42 47, 22. [6] R. Kumar and S. Takai. Decenralized rognosis of failures in discree even sysems. IEEE Trans. Auom. Conr., 55(:48 59, 2. [7] D. Lefebvre. Faul diagnosis and rognosis wih arially observed Peri nes. IEEE Trans. S.M.C.: Sys., 44(:43 424, 24. [8] A. Madalinski, F. ouioua, and P. Dague. Diagnosabiliy verificaion wih Peri ne unfoldings. In. J. Knowledge-Based and Inelligen Engineering Sys., 4(2:49 55, 2. [9] F. ouioua, P. Dague, and L. Ye. Predicabiliy in robabilisic discree even sysems. In Sof Mehods for Daa Sci., ages [2]. Ran, H. Su, A. Giua, and C. Seazu. Codiagnosabiliy analysis of bounded Peri nes. IEEE Transacions on Auomaic Conrol, 27. [2] S. Takai. Robus rognosabiliy for a se of arially observed discree even sysems. Auomaica, 5:23 3, 25. [22] S. Takai and R. Kumar. Inference-based decenralized rognosis in discree even sysems. IEEE Transacions on Auomaic Conrol, 56(:65 7, 2. [23] S. Takai and R. Kumar. Disribued failure rognosis of discree even sysems wih bounded-delay communicaions. IEEE Trans. Auom. Conr., 57(5: , 22. [24] L. Ye, P. Dague, and F. ouioua. Predicabiliy analysis of disribued discree even sysems. In 52nd CDC, ages 59 55, 23. [25] L. Ye, P. Dague, and F. ouioua. A redicabiliy algorihm for disribued discree even sysems. In Inernaional Conference on Formal Engineering Mehods, ages Sringer, 25. [26] H.-C. Yen. A unified aroach for deciding he exisence of cerain Peri ne ahs. Inform. Comuaion, 96(:9 37, 992. [27] X. Yin. Verificaion of rognosabiliy for labeled eri nes. IEEE Transacions on Auomaic Conrol, 28. [28] X. Yin and S. Laforune. A general aroach for solving dynamic sensor acivaion roblems for a class of roeries. In 54h IEEE Conference on Decision and Conrol, ages , 25. [29] X. Yin and S. Laforune. On he decidabiliy and comlexiy of diagnosabiliy for labeled Peri nes. IEEE Trans. Auom. Conr., 27. [3] X. Yin and Z. Li. Decenralized faul rognosis of discree-even sysems using sae-esimae-based roocols. IEEE Transacions on Cyberneics, 28. DOI: TCYB [3] X. Yin and Z.-J. Li. Decenralized faul rognosis of discree even sysems wih guaraneed erformance bound. Auomaica, 69: , 26. [32] X. Yin and Z.-J. Li. Reliable decenralized faul rognosis of discreeeven sysems. IEEE Trans. S.M.C.: Sys., 49(,