Minimization of Sensor Activation in Decentralized Discrete Event Systems

Transcription

1 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl Minimizatin f Sensr ctivatin in Decentralized Discrete Event Systems Xiang Yin, Memer, IEEE, Stéphane Lafrtune, Fellw, IEEE stract We investigate the prlem f dynamic sensr activatin fr decentralized decisin-making in partially-served discrete event systems, where the system is mnitred y a set f agents. The sensrs f each agent can e turned n/ff nline dynamically accrding t a sensr activatin plicy. We define a general decentralized decisin-making prlem called the decentralized state disamiguatin prlem, which cvers the decentralized cntrl prlem, the decentralized fault diagnsis prlem and the decentralized fault prgnsis prlem. The gal is t find a language-ased minimal sensr activatin plicy fr each agent such that the agents can always make a crrect glal decisin as a team. nvel apprach t slve this prlem is prpsed. We adpt a persn-y-persn apprach t decmpse this decentralized minimizatin prlem int tw centralized cnstrained minimizatin prlems. Each centralized cnstrained minimizatin prlem is then reduced t a fully centralized sensr activatin prlem that is slved effectively in the literature. The slutin tained is prvaly language-ased minimal with respect t the system language. Index Terms Discrete Event Systems, Dynamic Sensr ctivatin, Decentralized Decisin-Making. I. INTRODUCTION Decisin-making under limited sensr capacities is an imprtant prlem in netwrked autmated systems. Fr example, in the fault diagnsis prlem, the diagnsis mdule needs t infer the ccurrence f faults ased n its servatins. In this paper, we investigate the decisin-making prlem in Discrete Event Systems (DES) that perate under dynamic servatins. In this cntext, the system makes servatins nline thrugh its sensrs; these sensrs can e turned n/ff dynamically during the evlutin f the system accrding t a sensr activatin plicy that depends n the system s servatins. Due t energy, andwidth, r security cnstraints, sensrs activatins are cstly. Therefre, in rder t reduce sensr-related csts, it is f interest t minimize, in sme technical sense, the sensr activatins while maintaining sme desired servatinal prperty. In many large scale systems, the infrmatin structure is decentralized due t the distriuted nature f the sensrs. Under the decentralized setting, the system is mnitred y a set f agents that make lcal decisins and send them t a crdinatr. Then the crdinatr makes a glal decisin ased n the lcal decisins received frm each lcal agent. This wrk was partially supprted y the US Natinal Science Fundatin grants CCF (Expeditins in Cmputing prject ExCPE: Expeditins in Cmputer ugmented Prgram Engineering) and CNS X. Yin is with the Department f utmatin, Shanghai Jia Tng University, Shanghai , China ( xiangyin@sjtu.edu.cn). S. Lafrtune is with the Department f Electrical Engineering and Cmputer Science, University f Michigan, nn rr, MI 4809, US ( stephane@umich.edu). This decentralized decisin-making architecture is depicted in Figure. In the cntext f DES, many different decentralized decisin-making prlems have already een studied. Fr example, in the decentralized supervisry cntrl prlem [20], [4], lcal supervisrs need t disale/enale events dynamically in rder t restrict the system such that sme clsed-lp ehavir can e achieved. In the decentralized fault diagnsis prlem cnsidered in [7], [7], [34], the system is mnitred y a set f lcal agents that wrk as a team in rder t diagnse every ccurrence f fault events. Similarly, in the decentralized fault prgnsis prlem [5], the lcal agents need t wrk as a team in rder t predict every ccurrence f fault events. The prlem f ptimal sensr selectin fr static servatins has een widely studied in the DES literature; see, e.g., [], [0], [3], [42]. The gal in these wrks was t find an ptimal set f servale events that is fixed fr the entire executin f the system and enfrces a given DES-theretic prperty. In the cntext f dynamic servatins, where sensrs can e turned n/ff dynamically, the crrespnding prlem f ptimal sensr activatin has als received a lt f attentin in the literature; see, e.g., [4], [6], [22], [25], [27], [3], [33], [36], [37] fr a sample f this wrk and the recent survey paper [24] fr an extensive iligraphy. Fr example, in [4], [6], [27], [3] the prlem f centralized dynamic sensr activatin fr enfrcement f different diagnsaility prperties was slved. In [27], [33], dynamic sensr activatin fr the purpse f centralized cntrl was als studied. Recently, a general framewrk that slves a class f centralized dynamic sensr activatin prlems was prpsed [37]. Hwever, fr the decentralized sensr activatin prlem, there are very few results in the literature. In [3], the prlem f dynamic sensr activatin fr decentralized diagnsis is studied. Specifically, a windw-ased partitin apprach is prpsed in rder t tain a slutin. The main drawack f this apprach is that the slutin tained is nly ptimal with respect t a finite (restricted) slutin space and may nt e language-ased ptimal in general, where language-ased ptimal means that n sensr activatin plicy defined ver the entire language dmain can e strictly smaller (in terms f set inclusin) than the synthesized slutin. In ther wrds, y enlarging the slutin space y refining the state space f the system mdel, slutins etter than the slutin fund in [3] culd e tained in principle. Similarly, in [33], the prlem f dynamic sensr activatin fr decentralized cntrl is als studied, where the slutin tained is again ptimal w.r.t. a finite slutin space. T the est f ur knwledge, the prlem f language-ased sensr ptimizatin fr decentral-

2 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl ized diagnsis r cntrl has remained an pen prlem, as is mentined in the recent survey [24]. One imprtant reasn fr the lack f results fr the decentralized sensr activatin prlem is that decentralized multiplayer decisin prlems are cnceptually much mre difficult t slve than their crrespnding centralized versins. In particular, t synthesize a strategy fr ne agent, we need t knw the strategies f the ther agents, which are t e synthesized and again depend n the unknwn strategy f the first agent. In general, these types f prlems have een discussed in the framewrk f team decisin thery []. In the DES literature, it is well-knwn that many prlems that are decidale in the centralized setting ecme undecidale (e.g., the prlem f synthesizing safe and nn-lcking supervisrs [26], [28]) r pen (e.g., the prlem f synthesizing maximally permissive safe supervisrs [6]) in the decentralized case, even when nly tw agents are invlved. In this paper, we prpse a new apprach t tackle the prlem f dynamic sensr activatin fr the purpse f decentralized decisin-making. The main cntriutins f this paper are as fllws. First, we frmulate a general class f decentralized decisin-making prlems called the decentralized state disamiguatin prlem. We prpse the ntin f decentralized distinguishaility, which cvers cservaility, K-cdiagnsaility and cprgnsaility. Secnd, t slve the dynamic sensr activatin prlem, we adpt a persn-ypersn apprach (see, e.g., [30] and the references therein) t decmpse the decentralized minimizatin prlem t t- w cnsecutive centralized minimizatin prlems. We first minimize the sensr activatin plicy fr gent y keeping the plicy f gent 2 fixed. Then, we fix gent s sensr activatin plicy t the ne tained and slve the same minimizatin prlem ut fr gent 2. Essentially, we slve tw centralized cnstrained minimizatin prlems, since we need t take the ther agent s infrmatin int accunt when we minimize the decisins f an agent. nvel apprach is als prpsed t reduce each centralized cnstrained minimizatin prlem t a prlem that is slved effectively y an algrithm presented in [37]. Mrever, we prve that the slutin tained y ur prcedure is minimal with respect t the system language (i.e., ver an infinite set in general), in cntrast t the wrks reviewed ave where minimality was with respect t a finite slutin space. s special cases f the prpsed framewrk, language-ased sensr ptimizatins fr decentralized diagnsis and decentralized cntrl, which were previusly pen, are slved. These language-ased ptimal decentralized slutins essentially cme frm the effective reductin frm the decentralized prlem t tw fully centralized prlems and the language-ased ptimal slutin fund in [37] fr the fully centralized case. Finally, we shw that the prpsed framewrk is applicale t th the disjunctive architecture and the cnjunctive architecture. In general, a persn-y-persn apprach in team decisin prlems may nt terminate in a finite numer f steps, since we may need t iterate etween the tw cnstrained minimizatin prlems; see, e.g., [6]. Hwever, since we cnsider a lgical ptimality criterin, ur prlem enjys a mntnicity prperty, which decuples the minimizatin Ω D (s) gent P Ω D(s) Crdinatr Ω 2 D 2 (s) gent 2 P Ω2 Plant G Fig.. The decentralized decisin-making architecture with tw agents, where Ω i dentes gent i s sensr activatin plicy and P Ωi dentes the infrmatin mapping induced y Ω i. jective t sme extent. s a cnsequence, we can use the persn-y-persn apprach t effectively slve this prlem and iteratin is nt needed. Such a mntnicity prperty des nt hld in general fr aritrary decentralized synthesis prlems, e.g., [6], [26], [28]. In the DES literature, the persn-y-persn apprach has als een applied t many prlems, e.g., the decentralized cntrl prlem [6] and the decentralized cmmunicatin prlem [2], [8], [2]. In particular, in [3], [33], the persny-persn apprach is als explited fr slving the decentralized sensr activatin prlem fr the purpses f decentralized cntrl and decentralized diagnsis, respectively. The differences etween this paper and [3], [33] are as fllws. First, in [3] and [32], the authrs restrict the slutin spaces f the minimizatin prlems t finite dmains. Hwever, in general, the slutin space f the decentralized minimizatin prlem is infinite ver the system s language. Therefre, ne may find slutins that are etter than thse fund y [3], [33] ut are nt in the pre-specified finite slutin spaces. This infinite slutin space is als the fundamental difficulty in slving the decentralized minimizatin prlem. Hwever, ur apprach des nt need this restrictin, and cnsequently, the slutin tained in this paper is languageased minimal. Mrever, the prlem frmulatin in ur paper is mre general cmpared with thse in [3], [33]. Cnsequently, the results in this paper can e used t find language-ased minimal slutins fr a class f decentralized sensr activatin prlems under th the disjunctive and cnjunctive architectures, while the results in [3], [33] can nly e applied t specific prlems under the disjunctive architecture. Hwever, due t the unrestricted slutin space, ur algrithm has a higher cmplexity than thse in [3], [33]. The remainder f this paper is rganized as fllws. Sectin II descries the mdel f the system under dynamic servatins. In Sectin III, we frmulate the decentralized state disamiguatin prlem and the decentralized minimizatin prlem that we slve in this paper. Sectin IV shws hw t slve the centralized cnstrained minimizatin prlem y reducing it t a fully centralized prlem. In Sectin V, we present ur algrithm fr synthesizing a minimal decentralized slutin. In Sectin VI, we shw hw specific prlems, e.g., sensr activatin fr decentralized diagnsis/cntrl/prgnsis 0 s 2

3 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl can e slved y the prpsed framewrk. We als extend ur results t the cnjunctive architecture. Finally, we cnclude the paper in Sectin VII. Preliminary and partial versins f sme f the results in this paper are presented in [38]. Hwever, [38] nly investigated K-cdiagnsaility, which is a special case f the general framewrk prpsed in this paper. Mrever, the apprach used in this paper t slve the cnstraint minimizatin prlem is mre efficient than the state-partitin-autmatn-ased apprach in [38]. II. PRELIMINRIES ) System Mdel: Let Σ e a finite set f events. string is a finite sequence f events in Σ. We dente y Σ the set f all finite strings ver Σ including the empty string ɛ. Fr any string s Σ, we dente y s its length with ɛ = 0. language L Σ is a set f strings. We define L/s := {t Σ : st L} as the set f cntinuatins f string s in L. The prefix-clsure f language L Σ is L = {s Σ : w Σ s.t. sw L}. We say that L is prefix-clsed if L = L. We say that language L is live if s L, σ Σ : sσ L, i.e., any string in L can e extended t aritrarily lng length. We cnsider a DES mdeled as a deterministic finite-state autmatn (DF) G = (Q, Σ, δ, q 0 ), where Q is the finite set f states, Σ is the finite set f events, δ : Q Σ Q is the partial transitin functin and q 0 is the initial state. The functin δ is extended t Q Σ in the usual way (see, e.g., [3]). The ehavir f the system G starting frm state q Q is descried y the prefix-clsed language L(G, q) = {s Σ : δ(q, s)!}, where! means is defined. Fr the sake f simplicity, we als write δ(q, s) as δ(s) and write L(G, q) as L(G) if q = q 0. 2) Infrmatin Mapping: We cnsider a general dynamic servatin setting, where the servaility prperties f events can e cntrlled y a sensr activatin plicy during the evlutin f the system. Let Σ Σ e the set f events that can ecme servale y activating sme sensrs. sensr activatin plicy is defined as a pair Ω = (R, Θ), where R = (Q R, Σ, δ R, q 0,R ) is a deterministic autmatn such that L(R) = Σ and Θ : Q R 2 Σ is a laeling functin that specifies the current set f servale events within Σ. Specifically, fr any s Σ, Θ(δ R (s)) dentes the set f events that are mnitred after the ccurrence f s. While an event is mnitred, any ccurrence f it will e served. In ther wrds, after string s, events in Σ \ Θ(δ R (s)) are currently unservale (i.e., their sensrs are turned ff). Therefre, the cdmain f L, i.e., Σ, is the set f ptential servale events. T make Ω implementale, the pair (R, Θ) needs t satisfy the fllwing cnditins (C-) L(R) = Σ ; and (C-2) ( q, q Q R )( σ Σ:δ R (q, σ)=q )[q q σ Θ(q)]. The ave cnditins say that the sensing decisin can e updated (y updating the state f R) nly when a mnitred event ccurs. Mrever, Ω can react t any executin f the system as L(G) L(R) = Σ. In general, Q R culd e an infinite set. Hwever, we will shw later that the ptimal sensr activatin plicies f interest in this paper can always e cnstructed with finite state spaces. We say that the servatins are static if the set f servale events is fixed a priri. We dente y Ω Σ the crrespnding sensr activatin plicy fr the static servatin with the set f servale events Σ. Specifically, Ω Σ = (R, Θ) is given y: ) Q R = {q 0,R }; 2) σ Σ : δ R (q 0,R, σ) = q 0,R ; and 3) Θ(q 0,R ) = Σ. Given a sensr activatin plicy Ω = (R, Θ), we define the crrespnding infrmatin mapping P Ω : L(G) Σ recursively as fllws: { PΩ (s)σ if σ Θ(δ P Ω (ɛ) = ɛ, P Ω (sσ) = R (s)) P Ω (s) if σ Θ(δ R (s)) That is, P Ω (s) is the servatin f string s under Ω. Fr any language L Σ, we define P Ω (L) = {P Ω (s) Σ : s L}. Fr any tw sensr activatin plicies Ω=(R, Θ) and Ω = (R, Θ ), we write that Ω Ω if and write that Ω Ω if s L(G) : Θ (δ R (s)) Θ(δ R (s)) () [Ω Ω] [ s L(G):Θ (δ R (s)) Θ(δ R (s))] (2) 3) State Estimate: Let s L(G) e a string generated y the system. We dente y EΩ G (s) Q the state estimate upn the ccurrence f s w.r.t Ω and the state space f G. Specifically, fr any s L(G), we have E G Ω (s):={δ(t) Q: t L(G) s.t. P Ω (s)=p Ω (t)} Clearly, if P Ω (s) = P Ω (t), then EΩ G(s) = E Ω G (t). T cmpute EΩ G (s), we can cnstruct the server f G. Let Ω = (R, Θ), R = (Q R, Σ, δ R, q 0,R ) e a sensr activatin plicy. The server fr G under Ω is Os Ω (G) = (X, Σ, f, x 0 ), (3) where X 2 Q Q R is the state space and fr any state x X, we write x = (I(x), R(x)) where I(x) 2 Q and R(x) Q R. The partial transitin functin f the server f : X Σ X is defined as fllws: fr any x = (ı, q), x = (ı, q ) X and σ Θ(q) Σ, f(x, σ)=x iff { q = δ R (q, σ) ı (4) = UR Θ(q )(Next σ (ı)) where fr any ı 2 Q, σ Σ and θ 2 Σ, Next σ (ı) = {q Q : q 2 ı s.t. δ(q 2, σ)=q } UR θ (ı) = {q Q : q 2 ı, s (Σ \ θ) s.t. δ(q 2, s)=q } Intuitively, Next σ (ı) is the set f states that can e reached frm sme state in ı immediately after serving σ and UR θ (ı) is the set f states that can e reached unservaly frm sme state in ı under the set f mnitred events θ. Finally, the initial state f Os Ω (G) is x 0 = (UR Θ(q0,R )({q 0 }), q 0,R ). Then the state estimate EΩ G (s) can e cmputed y I(f(P Ω (s))) = EΩ G (s), i.e., the state cmpnents f the server state reached upn P Ω (s) is the state estimatr value after s. ls, if Ω Ω, then we have s L(G) : EΩ G(s) E Ω G (s) [6]. Example. Cnsider the system G in Figure 2(a). Let Σ, = {, a} and Σ,2 = {, } e tw sets f servale

4 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl *+ 2 3 *+ *a+ *+ *, + 2 Σ a 2 3 *+ *a+ f (,2,3,4,6 2,-,-, ) f,-,- 2 ( 3,3,5,7,-, 0, ) f f a 4 5 ( 5 Σ\ 4,, ) 0,- Σ\a a Σ ( 5,- 2,4,7 Σ\,, 2) Σ a a a Σ 2 3 a ( 7 6,,- 7,-, *, + *+, ) *a+ (*+ 6, 3) (a) System G () Ω (c) Os Ω (G) Σ *, + (d) Ω 2 (,3, ) (,3,5,7 Σ\ Σ\a ( 5, ) Σ Σ\ Σ a a ( 2,4,6, ) *+ *a+ ( 7, ) (e) Os Ω2 (G) *+ ( 2,4,7, ( 6, 3 Fig. 2. Examples f sensr activatin plicies (,2,3,4,6 and servers, ) (,3,5,7, ) ( 5, ) ( 2,4,7, 2) events. s shwn in Figure 2(), Ω is a sensr activatin a ( 7, ) ( 6, 3) plicy with the set f servale events Σ,. The laeling functin is specified y the set f events assciated with each state in the figure. Initially, event is mnitred y Ω. Once is served, Ω changes t mnitr event a. Finally, Ω turns all sensrs ff when a is served. The crrespnding server Os Ω (G) is shwn in Figure 2(c). Fr example, fr the string f a L(G), we have that P Ω (f a) = a and I(f(a)) = {6} = E G Ω (f a). Similarly, Figure 2() shws a sensr activatin plicy Ω 2 with the set f servale events Σ,2. Clearly, Ω 2 always mnitrs all events in Σ,2, i.e., Ω 2 = Ω Σ,2. Therefre, the server Os Ω2 (G) shwn in Figure 2(c) is the standard server (see, e.g, [3]) if we ignre the secnd cmpnent f each state. III. DECENTRLIZED STTE DISMIGUTION PROLEM In this sectin, we first define the ntin f decentralized distinguishaility. Then we frmulate the decentralized sensr minimizatin prlem fr the purpse f state disamiguatin.. Decentralized Distinguishaility In the decentralized decisin-making prlem, at each instant, each lcal agent sends highly cmpressed infrmatin, i.e., a lcal decisin, t the crdinatr ased n its lcal (dynamic) servatin. Then the crdinatr makes a glal decisin ased n the infrmatin received frm each lcal agent. Let I e the index set f lcal agents. Fr each agent i I, we dente y Ω i its sensr activatin plicy and y Σ,i the set f events that can e mnitred in Ω i. Fr the sake f simplicity, we develp all results hereafter fr the case f tw agents, i.e., I = {, 2}. The principle can e extended t an aritrary numer f agents. We define the pair f sensr activatin plicies as Ω = [Ω, Ω 2 ]. In rder t frmulate the decentralized decisin-making prlem, we need t specify the fllwing three ingredients: What requirement the glal decisin has t fulfill? What infrmatin each lcal agent can send t the crdinatr? What rule the crdinatr uses t calculate a glal decisin ased n the lcal decisins? Hereafter, we refer t the first ingredient as the specificatin f the decentralized decisin-making prlem. The last tw ingredients are referred t as the architecture f the decentralized decisin-making prlem. (,2,3,4,6, ) (,3,5,7, ) ( 5, ) ( 2,4,7, 2) a ( 7, ) ( 6, 3) Several different specificatins have een studied separately in the literature fr decentralized decisin-making prlems, e.g., t diagnse every ccurrence f fault events [7], [7], t predict every ccurrence f fault events [5] r t cntrl the system [20], [4]. In this paper, we d nt study a specific specificatin. Instead, we define a general class f specificatins called decentralized state disamiguatin. s shwn later in Sectin VI that many existing decentralized decisinmaking prlems are special cases f the decentralized disamiguatin prlem. Frmally, we define a specificatin as a pair f state sets T = Q T Q T Q Q (5) Intuitively, specificatin T is used t capture the fllwing requirement. State set Q T represents the set f states at which the glal system must take sme desired actin assciated t T and state set Q T represents the set f states at which the glal system shuld nt take such an actin. Then the system must e ale t distinguish etween states in Q T and states in Q T (under certain decentralized architecture, which will e specified later) when a state in Q T is reached; therwise, the desired actin assciated t T cannt e taken safely. Regarding the architecture f the decentralized decisinmaking prlem, here we cnsider the fllwing mechanism, which is widely used in the literature fr many different prlems [7], [5], [7], [20], [4]. We assume that cmmunicatin etween each agent and the crdinatr is cstly and nly a inary decisin is allwed fr each agent at each instant. That is, each lcal agent can nly send t the crdinatr a highly cmpressed decisin r 0, which crrespnd t take the actin and d nt take the actin, respectively. Then, the crdinatr has tw pssile fusin rules t tain a glal decisin frm lcal decisins: the disjunctive rule: issues glally if and nly if ne lcal agent issues. the cnjunctive rule: issues glally if and nly if all lcal agents issue. Hereafter, we will develp the main results ased n the disjunctive rule. We will discuss hw t extend ur results t the cnjunctive case in Sectin VI-D. In general, the system may have multiple distinct jectives, i.e., it needs t distinguish different states pairs fr different purpses. Fr the sake f generality, we cnsider m specificatins and dente y T = {T,..., T m } the set f specificatins, where T k = QT k Q Q, T k T. ls, fr the sake f generality, fr each T k T, we define

5 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl I Tk I as the nn-empty set f agents that can cntriute t the decisin assciated t T k. If I Tk is a singletn, then the glal decisin will e if the unique agent in I Tk issues. Hwever, in the case that I Tk >, since we cnsider the disjunctive architecture, the glal decisin will e if ne agent in I Tk issues. Therefre, an agent must e ale t distinguish any states pair in T k unamiguusly when it issues ; therwise a wrng glal decisin may e made. This servatin leads t the fllwing definitin f decentralized distinguishaility. Definitin. (Decentralized Distinguishaility). Let G e the system, T = {T,..., T m } e a set f specificatins and Ω = [Ω, Ω 2 ] e a pair f sensr activatin plicies. We say that G is decentralized distinguishale w.r.t. Ω and T if ( T k T)( s L(G):δ(s) )( i I T k )[E G Ω i (s) = ] (6) Intuitively, the ave definitin says the fllwing. Fr any specificatin T k T, fr any string that ges t a state in, i.e., a state at which we must take the actin assciated t T k, there must exist at least ne lcal agent in I Tk that knws fr sure that we can take such an actin. Nte that, in ur setting, nly are the set f states at which we cannt take the actin assciated t T k. In ther wrds, there is n harm in taking the actin if the system is in Q \ ( QT k ). This is why we require E Ω G i (s) = rather than EΩ G i (s). We will shw later in Sectin VI that K-cdiagnsaility, cservaility and cprgnsaility are all instances f decentralized distinguishaility. Nte that, if QT k fr sme T k T, then G will nt e decentralized distinguishale fr any sensr activatin plicies Ω. This phenmenn may ccur in the fault prgnsis prlem as we will discuss later in Sectin VI-C. Example 2. We still cnsider the system G in Figure 2(a) and Σ, = {, a} and Σ,2 = {, } are tw sets f servale events. We assume that the servatins are static, i.e., Ω = Ω Σ, and Ω 2 = Ω Σ,2. Let us cnsider the fllwing set f specificatins T = {T, T 2 }, where T = Q T T 2 = Q T2 QT QT2 = {6} {, 2, 3, 5, 7} = {5, 7} {, 2, 4, 6} and I T = I T2 = {, 2}. We can verify that G is decentralized distinguishale w.r.t. {T, T 2 } and [Ω Σ,, Ω Σ,2 ]. Fr example, fr specificatin T and string f a such that δ(f a) = 6 Q T, we have I T and EΩ G Σ, (f a) Q T = {6} {, 2, 3, 5, 7} =. Hwever, if we add anther specificatin T 3 = {4} {, 2} t {T, T 2 }, then G will nt e decentralized distinguishale. Fr example, fr δ(f ) = 4 Q T3, we have E Ω G Σ, (f ) Q T3 = {2, 4} {, 2} and EΩ G Σ,2 (f ) Q T3 = {2, 4, 6} {, 2}, i.e., nne f the agents can distinguish specificatin T 3. Remark. The state disamiguatin prlem and its sensr activatin have een studied in the literature in the centralized setting; see, e.g., [6], [23], [32]. Cmpared t its centralized cunterpart, the decentralized disamiguatin prlem has the fllwing imprtant difference. In the centralized setting, specificatin Q Q and specificatin Q Q are equivalent in the sense that if the system can distinguish state q frm state q 2, then it can als distinguish q 2 frm q. Hwever, it is nt the case in the decentralized setting and we cannt swap Q and Q aritrarily. One can easily verify that G is decentralized distinguishale w.r.t. Q Q des nt necessarily imply that it is decentralized distinguishale w.r.t. Q Q. Mrever, ur prcedure fr slving the sensr activatin prlem in the decentralized setting is cmpletely different frm thse in the centralized case.. Prlem Frmulatin and Slutin Overview Let T e the set f specificatins. Then the gal f the sensr activatin prlem is t find an ptimal pair f sensr activatin plicies Ω = [Ω, Ω 2 ] such that the system is decentralized distinguishale w.r.t. Ω and T. In this paper, we cnsider the lgical ptimality criterin that is widely used in the literature [24], [3], [33]. Specifically, fr any Ω = [Ω, Ω 2 ] and Ω = [Ω, Ω 2], the inclusin Ω Ω means that i I : Ω i Ω i (7) and the strict inclusin Ω Ω means that [ Ω Ω] [ i I : Ω i Ω i ] (8) We are nw ready t frmulate the prlem f minimal sensr activatin fr decentralized state disamiguatin. Prlem. Let G e the system and T = {T,..., T m } e a set f specificatins. Fr each agent i {, 2}, let Σ,i Σ e the set f servale events. Find sensr activatin plicies Ω = [Ω, Ω 2] such that: C. G is decentralized distinguishale w.r.t. Ω and T. C2. Ω is minimal, i.e., there des nt exist anther Ω Ω that satisfies (C). Remark 2. In [3], [33], su-ptimal slutins t tw special cases f Prlem, the decentralized cntrl prlem and the decentralized diagnsis prlem, are prvided, in the sense that the slutins fund therein are minimal amng all slutins ver given finite restricted slutin spaces. In principle, the slutins fund in [3], [33] culd e imprved y emplying finer partitins and repeating the ptimizatin prcedure. In this paper, we are aiming fr a language-ased minimal slutin, in the sense that the ntin f strict inclusin f sensr activatin plicies is defined in terms f the strings in L(G) (see Equatins (2) and (8)). In ther wrds, we d nt impse, a priri, any cnstraints n the slutin space f each Ω i. Hence, n etter slutin can e tained y refining the state space f G and repeating the slutin prcedure. T the est f ur knwledge, such a language-ased ptimal slutin t the decentralized sensr activatin prlem has never een reprted in the literature. Mrever, Prlem is mre general than the prlems studied in [3], [33]. efre we frmally tackle Prlem, let us first prvide a rief verview f ur slutin apprach. We adpt the persn-y-persn apprach that has een widely used in decentralized ptimizatin prlems. Specifically, we decmpse

6 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl the decentralized minimizatin prlem t a set f centralized cnstrained minimizatin prlems and fr each such prlem, we nly attempt t minimize ne agent s sensr activatin plicy while the ther ne is fixed. Hwever, the fllwing questins arise. First, y taking the persn-y-persn apprach, iteratins invlving minimizatin fr each agent may e required in general, and such iteratins may nt terminate in a finite numer f steps. We will shw that in ur particular prlem such iteratins are nt required. This is due f the s-called mntnicity prperty that arises in dynamic sensr activatin prlems. The secnd questin f interest is hw t minimize the sensr activatin plicy f ne agent when the plicy f the ther agent is fixed. This prlem is different frm the fully centralized minimizatin prlem, since we shuld nt nly cnsider the infrmatin f the agent whse sensr activatin plicy we are minimizing, ut we must als take int accunt the infrmatin availale t the ther agent, whse sensr activatin plicy is fixed. Therefre, the true infrmatin state fr this minimizatin prlem cnsist f (i) the knwledge f the agent whse sensr activatin plicy is eing minimized; and (ii) this agent s inference f the ther agent s ptential knwledge f the system ased n that agent s wn infrmatin. T reslve this infrmatin dependency, we develp a nvel apprach y which we encde the secnd agent s knwledge int the system mdel. This is discussed in the next sectin. IV. CONSTRINED MINIMIZTION PROLEM In this sectin, we tackle the prlem f minimizing the sensr activatin plicy fr ne agent when the sensr activatin plicy f the ther ne is fixed. This prlem is als referred t as the centralized cnstrained minimizatin prlem herafter. Thrughut this sectin, i {, 2} dentes the agent whse sensr activatin plicy is eing minimized while j {, 2}, j i dentes the ther agent whse sensr activatin plicy is fixed.. Cnstrained Minimizatin Prlem Prlem 2. (Centralized Cnstrained Minimizatin Prlem). Let G e the system and T = {T,..., T m } e a set f specificatins. Let i, j {, 2}, i j e tw agents. Suppse that the sensr activatin plicy Ω j fr gent j is fixed. Find a sensr activatin plicy Ω i fr gent i such that: C. G is decentralized distinguishale w.r.t. [Ω, Ω 2 ] and T. C2. Fr any Ω i satisfying (C), we have Ω i Ω i. The ave prlem is different frm th the centralized and decentralized minimizatin prlems. In the centralized minimizatin prlem, where nly ne agent is invlved, t maintain distinguishaility, we need t require that T k T, s L(G) : (E G Ω (s) E G Ω (s)) T k = where Ω is the centralized sensr activatin plicy. In ther wrds, the agent shuld always e ale t distinguish states in frm states in QT k fr any T k T. Hwever, in the decentralized disamiguatin prlem, it is pssile that there exists a string s L(G), δ(s) such that E Ω G i (s), ut EΩ G j (s) =, where j I T k. Therefre, gent j may help gent i t reslve the amiguity. In ther wrds, t slve the cnstrained minimizatin prlem fr ne agent, we must take the ther agent s sensr activatin plicy int accunt.. Prlem Reductin First, we recall a general class f fully centralized sensr activatin prlems that is studied in [37]. Prlem 3. (Centralized Sensr Minimizatin Prlem fr IS-ased Prperty). Let G = (Q, Σ, δ, q 0 ) e the system and ϕ : 2 Q {0, } e a functin. Find a sensr activatin plicy Ω such that C. s L(G) : ϕ(eω G (s)) = ; and C2. Fr any Ω satisfying (C), we have Ω Ω. Prlem 3 is a fully centralized sensr activatin prlem, since nly ne agent is invlved. In particular, functin ϕ : 2 Q {0, } is referred t as an infrmatin-state-ased prperty. This prlem is studied in mre detail in [37], where an algrithm is prvided that slves this prlem effectively y returning a finite sensr activatin plicy satisfying the requirements. Nte that, the algrithm in [37] als guarantees y cnstructin that the synthesized sensr activatin plicy satisfies the implementatin cnditins (C-) and (C-2). In general, a minimal sensr activatin plicy des nt exist fr an aritrary prperty, e.g., detectaility r diagnsaility withut a pre-specified delay. Hwever, fr an IS-ased prperty, a minimal sensr activatin plicy des exist and it is finitely realizale; this is ecause a IS-ased prperty can e checked ver the state estimates f the system. Therefre, if we can reduce Prlem 2 t Prlem 3, then it means that Prlem 2 can als e slved effectively and the slutin will e finitely realizale. We nw shw that such a reductin is pssile y using autmata V and Ṽ, which are defined next. Let G e the system and Ω j e the fixed sensr activatin plicy, where Ω j = (R j, Θ j ) and R j = (Q j R, Σ, δj R, qj 0,R ). We define a new autmatn V = (Q V, δ V, Σ V, q 0,V ) (9) where Q V Q Q j R Q Qj R is the set f states; Σ V = (Σ {ɛ}) (Σ {ɛ}) is the set f events; q 0,V = (q 0, q j 0,R, q 0, q j 0,R ) is the initial state; The transitin functin δ V : Q V Σ V Q V is defined y: fr any (q, q R, q 2, q2 R ) and σ Σ, the fllwing transitins are defined: If σ Θ j (q R ) and σ Θ j (q2 R ), then δ V ((q, q R, q 2, q R 2 ), (σ, σ)) =(δ(q, σ), δ j R (qr, σ), δ(q 2, σ), δ j R (qr 2, σ)) If σ Θ j (q R ) and σ / Θ j (q R 2 ), then δ V ((q, q R, q 2, q R 2 ), (ɛ, σ)) = (q, q R, δ(q 2, σ), δ j R (qr 2, σ)) If σ / Θ j (q R ) and σ Θ j (q R 2 ), then δ V ((q, q R, q 2, q R 2 ), (σ, ɛ)) = (δ(q, σ), δ j R (qr, σ), q 2, q R 2 )

7 *, + *+ *a+ *+ This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl (,3, ) ( 2,4,6, ) ( 5, ) If σ / Θ j (q R ) and σ / Θ j (q2 R ), then δ V ((q, q R, q 2, q2 R ), (σ, ɛ)) = (δ(q, σ), δ j ( 7 R (qr, σ), q 2, q2 R, ) ) δ V ((q, q R, q 2, q R 2 ), (ɛ, σ)) = (q, q R, δ(q 2, σ), δ j R (qr 2, σ)) The ave cnstructin fllws the well-knwn (,3,5,7 M-machine, ) (r twin-plant) cnstructin that was riginally used fr the verificatin f (c)servaility [2], [9], [29]; ( 2,4,7 ut, we 2) generalize it t the dynamic servatin setting. Essentially, a V tracks a pair f strings that lk the same fr gent ( 6 j, under 3) Ω j. Specifically, the first tw cmpnents are used t track a string in the riginal system and the last tw cmpnents are used t track a string that lks the same as the first string. Since we are cnsidering the dynamic servatin setting, we als need t track states in the sensr activatin plicy in rder t determine the set f mnitred events; this is why the secnd (respectively, furth) cmpnent always mves tgether with the first (respectively, third) cmpnent. Therefre, fr any (s, s 2 ) L(V ), we have that P Ωj (s ) = P Ωj (s 2 ). Similarly, fr any t, w L(G) such that P Ωj (t) = P Ωj (w), there exists (s, s 2 ) L(V ) such that s = t and s 2 = w, i.e., state (δ(t), δ j R (t), δ(w), δj R (w)) is reachale in V. Next, we mdify V as fllws. Fr each transitin in V, if the event is in the frm f (σ, σ) r (σ, ɛ), then we replace the event y σ; if the event is in the frm f (ɛ, σ), then we replace the event y ɛ. We dente y Ṽ = (Q Ṽ, δ Ṽ, Σ Ṽ, q 0,Ṽ ) the mdified autmatn. Similar mdificatin was als used in [40], [43] in the static servatin setting fr different purpse. Intuitively, Ṽ nly keeps the first cmpnent f the event f each transitin in V, since this part crrespnds t the transitin in the real system. Nte that Ṽ is a nn-deterministic autmatn, since ɛ-transitin is allwed. Therefre, δṽ (s) is the set f states that can e reached frm q 0, Ṽ via s. The mdified autmatn Ṽ has the fllwing prperties. First, we have that L(Ṽ ) = L(G). Clearly, L(Ṽ ) L(G) since a transitin in Ṽ is defined nly when the crrespnding transitin in G is defined. ls, fr any string s L(G), we knw that (s, s) L(V ), which implies that s L(Ṽ ). Secnd, fr any s L(Ṽ ) = L(G), we knw that δṽ (s) (0) ={(δ(s), δ j R (s), δ(t), δj R (t)) Q Ṽ : (s, t) L(V )} ={(δ(s), δ j R (s), δ(t), δj R (t)) Q Ṽ :t L(G) P Ω j (s)=p Ωj (t)} Therefre, fr any string s L(G) = L(Ṽ ), if (q, q R, q 2, q2 R ) δṽ (s), then it implies that δ(s) = q and state q 2 cannt e distinguished frm q under Ω j. Fr any x 2 QṼ, we dente y I (x) = {q Q : (q, q R, q 2, q2 R ) x} the set f states in the first cmpnent f x. Then, fr any sensr activatin plicy Ω, y Equatin (0), we have EΩ G(s) = I (E ṼΩ (s)) fr any s L(G). Example 3. Let us still cnsider the system G shwn in Figure 2(a). Suppse that the fixed Ω j is the sensr activatin plicy Ω 2 shwn in Figure 2(d), i.e., Ω j always mnitrs and. Then autmatn Ṽ cnstructed frm G and Ω j is shwn in Figure 3. Clearly, we see that L(Ṽ ) = L(G). Fr (2,,2,) (,,,) f ε f 2 ε (4,,2,) (2,,4,) (3,,,) (,,3,) a ε f ε ε f 2 (6,,2,) (4,,4,) (2,,6,) (3,,3,) ε a ε f (6,,4,) (4,,6,) (5,,5,) ε a (6,,6,) (7,,7,) Fig. 3. utmatn Ṽ. string f a L(Ṽ ) = L(G), we have that δ Ṽ (f a) = {(6,, 2, ), (6,, 4, ), (6,, 6, )} and I (E ṼΩ j (f a)) = I ({(2,, 2, ), (4,, 2, ), (6,, 2, ), (2,, 4, ), (4,, 4, ), (2,, 6, ), (4,, 6, ), (6,, 6, )}) = {2, 4, 6} = E G Ω j (f a). Nw, let us shw hw t use Ṽ t reduce the cnstraint minimizatin prlem, i.e., Prlem 2, t a fully centralized minimizatin prlem, i.e., Prlem 3. First, we define the distinguishaility functin DF : 2 QṼ {0, } as fllws: fr each x 2 QṼ, {, if Tk T : (c-i) r (c-ii) hlds DF (x)= () 0, therwise where cnditins (c-i) and (c-ii) are defined y: (c-i) i I Tk and (I (x) I (x)) T k =. (c-ii) j I Tk and q I (x), (q, q R, q 2, q2 R ) x : (q, q 2 ) / T k. Let us explain the intuitin f the ave tw cnditins in functin DF. Suppse that Ω i is the sensr activatin plicy t e synthesized fr gent i. Let s L(G) e a string such that δ(s), i.e., the crdinatr must take the actin assciated t T k when s is executed. Then E ṼΩ i (s) is the state estimate w.r.t. the state space f Ṽ under Ω i. Essentially, functin DF evaluates whether r nt decentralized distinguishaility is fulfilled y checking whether r nt x := E ṼΩ i (s) satisfies cnditins (c-i) and (c-ii), which can e interpreted as fllws. - If (c-i) hlds, then we knw that gent i can cntriute t the glal decisin assciated t T k, since i I Tk. Mrever, it can cntriute t the right decisin since it knws fr sure that the actin assciated t T k has t e taken, since (EΩ G i (s) EΩ G i (s)) T k =. Therefre, the disamiguatin requirement is fulfilled even withut lking at gent j. - If (c-i) des nt hld, then we knw that either gent i cannt cntriute t t the glal decisin assciated t T k r gent i cannt make a right decisin due t states amiguity, i.e., q, q 2 EΩ G i (s) = I (E ṼΩ i (s)) : (q, q 2 ) T k. In rder t issue the right glal decisin, gent j must e ale t help gent i t distinguish thse amiguus strings, i.e., cnditin (c-ii) needs t hld. First, gent j shuld e ale t cntriute t the glal decisin assciated t T k, i.e., j I Tk. Then, fr any string t that lks the same as s fr gent i and leads t a state in, there shuld nt exist anther string w that lks the same as t fr gent j and leads t a state in. Recall that Ṽ is cnstructed

8 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl y tracking all states that cannt e distinguished frm q y gent j. Therefre, gent i can infer which states gent j cannt distinguish y using Ṽ. Specifically, if fr any (q, q R, q 2, q2 R ) x : (q, q 2 ) / T k, then we knw that there is n such a string w that can cnfuse gent j fr sme string t, i.e., gent j can make a right decisin assciated t T k. Finally, we wuld like t remark that, althugh specificatin T is defined ver the state space f G, the distinguishaility functin DF is defined ver the state space f Ṽ, i.e., we need t slve Prlem 3 fr the mdified system Ṽ. Hwever, this is nt a prlem, since the first cmpnent f a state Ṽ exactly carries the same state infrmatin in G, i.e., I (E ṼΩ i (s)) = EΩ G i (s) fr any Ω i. Mrever, since L(Ṽ ) = L(G), we knw that Ṽ and G have the same servale ehavir under any sensr activatin plicy. Therefre, we can first use Ṽ t synthesize a sensr activatin plicy and then use it t mnitr G. We summarize the ave discussins y the fllwing therem. Therem. Let G e the system and T = {T,..., T m } e a set f specificatins. Let Ṽ e the autmatn cnstructed ased n Ω j. Then, G is decentralized distinguishale w.r.t. [Ω, Ω 2 ] and T if and nly if s L(G) : DF (E ṼΩ i (s)) = (2) Prf. ( ) y cntrapsitin. Suppse that L(G) is nt decentralized distinguishale. Then, there exists T k T, such that ( s L(G):δ(s) )( p I T k )[EΩ G p (s) ] (3) Let us cnsider the fllwing three cases fr I Tk. Case : I Tk = {i}. Let us cnsider (c-i), since (c-ii) is vilated directly. y Equatin (3), since δ(s) E Ω G i (s) and E Ω G i (s), we have (I (E ṼΩ i (s)) I (E ṼΩ i (s))) T k =(E G Ω i (s) E G Ω i (s)) T k Therefre, (c-i) is als vilated and DF (E ṼΩ i (s)) = 0. Case 2: I Tk = {j}. Let us cnsider (c-ii), since (c-i) is vilated directly. We still cnsider string s in Equatin (3). We have δ(s) EΩ G i (s) = I (E ṼΩ i (s)). Since E Ω G j (s), there exists a string t L(G) such that P Ωj (s) = P Ωj (t) and δ(t). This implies that (δ(s), δ(t)) T k. Since P Ωj (s) = P Ωj (t), y the cnstructin f Ṽ, (δ(s), δj R (s), δ(t), δj R (t)) δ Ṽ (s) E ṼΩ i (s). Therefre, (c-ii) is als vilated and DF (E ṼΩ i (s)) = 0. Case 3: I Tk = {, 2}. Fr string s in Equatin (3), since EΩ G i (s), y the same argument as in Case, (c-i) des nt hld. Since EΩ G j (s), y the same argument as in Case 2, (c-ii) als des nt hld. Therefre, DF (E ṼΩ i (s)) = 0. Overall, fr each case, there exists a string s L(G) such that DF (E ṼΩ i (s)) = 0, which cmpletes the cntrapsitive prf. ( ) Still y cntrapsitive. Suppse that there exists a string s L(G) such that DF (E ṼΩ i (s)) = 0. Then, there exists T k T such that nne f (c-i) and (c-ii) hlds fr E ṼΩ i (s). Next, we still cnsider the fllwing three cases fr I Tk. Case : I Tk = {i}. Since (c-i) des nt hld, we have (EΩ G i (s) EΩ G i (s)) T k = (I (E ṼΩ i (s)) I (E ṼΩ i (s))) T k. This implies that w L(G) such that δ(w), P Ω i (s) = P Ωi (w) and EΩ G i (w) = E Ω G i (s). Therefre, we have ( w L(G) : δ(w) )[E Ω G i (w) ], i.e., G is nt decentralized distinguishale. Case 2: I Tk = {j}. Since (c-ii) des nt hld, we have q E G Ω i (s), (q, q R, q 2, q R 2 ) E ṼΩ i (s) : (q, q 2 ) T k Since (q, q R, q 2, q2 R ) E ṼΩ i (s), there exists a string t L(Ṽ ) = L(G), such that P Ω i (s) = P Ωi (t) and (q, q R, q 2, q2 R ) δṽ (t), which further implies that q = δ(t) and there exists w L(G) such that q 2 = δ(w) and P Ωj (t) = P Ωj (w). Therefre, {q, q 2 } EΩ G j (t) = EΩ G j (w). Since (q, q 2 ) T k, we knw that q and q 2. Overall, fr T k T, we have ( t L(G):δ(t) )[E Ω G j (t) ], i.e., G is nt decentralized distinguishale. Case 3: I Tk = {, 2}. Since (c-ii) des nt hld, y the same argument as in Case 2, there exists t L(G) such that P Ωi (s) = P Ωi (t), δ(t) and EΩ G j (t). Since (c-i) des nt hld, y the same argument as in Case, we knw that E G Ω i (s) P Ωi (s) = P Ωi (t), we have E G Ω i (t). Since. Therefre, ( t L(G):δ(t) )( p I T k )[E G Ω p (t) ] i.e., G is nt decentralized distinguishale. Overall, G is nt decentralized distinguishale fr each case. This cmpletes the cntrapsitive prf. In the ave develpment, the essence f using Ṽ is that we can encde gent j s infrmatin, i.e., Ω j, int the system mdel in rder t reduce the cnstrained minimizatin prlem fr gent i t a fully centralized minimizatin prlem. That is, Ṽ is a nn-deterministic refinement f G that carries th the riginal state infrmatin in G and sme useful infrmatin f Ω j. Once Ṽ is cnstructed, we will nt use Ω j anymre, since all useful infrmatin, i.e., which pairs f states gent j cannt distinguish, has een encded in Ṽ. Finally, using Therem, we have the fllwing result. Crllary. Prlem 2 is decidale. Prf. y Therem, it is clear that Prlem 2 is a special case f Prlem 3 y cnsidering system Ṽ and setting ϕ t e DF : 2 QṼ {0, }. Since Prlem 3 can e effectively slved, Prlem 2 can als e effectively slved. Example 4. We return t the system G in Figure 2(a) with Σ, = {, a} and Σ,2 = {, }. We still cnsider specificatins T = {T, T 2 } defined in Example 2. We assume that sensr activatin plicy Ω 2 shwn in Figure 2(d) is fixed fr gent 2 and the crrespnding autmatn Ṽ has een

9 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl shwn in Figure 3. Nw, we want t synthesize sensr activatin plicy Ω such that G is decentralized distinguishale. y defining functin DF fr Ṽ and applying the synthesis algrithm in [37], we tain a minimal sensr activatin plicy Ω shwn in Figure 4(a). Since the main purpse f this paper is t shw hw t slve the decentralized minimizatin prlem, the reader is referred t [37] fr mre details aut the slutin apprach t Prlem 3. Here, instead f shwing hw t find Ω, let us verify that Ω satisfies functin DF. Fr example, fr specificatin T, we cnsider string f a such that δ(f a) = 6. Then we have x = E ṼΩ (f a) = {(6,, 6, )}, i.e., I (x) = {6}. Therefre, cnditin (c-i) hlds fr x and we have DF (x) =. Fr specificatin T 2, let us cnsider string f 2 such that δ(f 2 ) = 5 Q T2. Then we have x = E ṼΩ (f 2) = {(,,, ), (3,,,, ), (,, 3, ), (3,, 3, ), (5,, 5, ), (7,, 7, )}, i.e., I (x) = {, 3, 5, 7}. Fr this case, cnditin (c-i) des nt hld fr x since I (x) Q T2. Hwever, fr 5 I (x) Q T2 = {5, 7}, (5,, 5, ) is the nly state in x whse first cmpnent is 5 and (5, 5) / T 2. Similarly, fr 7 I (x) Q T2, (7,, 7, ) is the nly state in x whse first cmpnent is 7 and (7, 7) / T 2. Therefre, cnditin (c-ii) hlds and we still have DF (x) =. V. SYNTHESIS LGORITHM In this sectin, we first present an algrithm that slves the decentralized sensr activatin prlem y using the results we develped s far. Then we prve the crrectness f the algrithm. Our synthesis algrithm is frmally presented in lgrithm D-MIN-CT. Essentially, lgrithm D-MIN-CT slves tw centralized cnstrained minimizatin prlems. First, we set gent 2 s sensr activatin plicy t e Ω Σ,2, i.e., the mst cnservative ne, and slve the cnstrained minimizatin prlem fr gent. Then we fix the tained sensr activatin plicy fr gent and slve the cnstrained minimizatin prlem fr gent 2. Hwever, the fllwing questin arises: fter the ave prcedure, d we need t fix gent 2 s new sensr activatin plicy and g ack t minimize gent s sensr activatin plicy again? In ther wrds, we need t answer whether r nt iteratins etween tw centralized cnstrained minimizatin prlems are required in rder t tain a decentralized minimal slutin. Hereafter, we shw that such iteratins are nt necessary fr ur prlem and lgrithm D-MIN-CT indeed yields a decentralized minimal slutin in the ave tw steps. This is ecause f the fllwing mntnicity prperty which generalizes the results in [3], [33]. Lemma. (Mntnicity Prperty). Let G e the system, T e a set f specificatins and Ω = [Ω, Ω 2 ] and Ω = [Ω, Ω 2] e tw sensr activatin plicies such that Ω Ω. Then G is decentralized distinguishale w.r.t. Ω and T implies that G is decentralized distinguishale w.r.t. Ω and T. Prf. Since G is decentralized distinguishale w.r.t. Ω and T, then T k T, s L(G) : δ(s), we have i {, 2} : EΩ G Q i(s) T k =. Since Ω Ω, we knw that lgrithm : D-MIN-CT input : G, T, Σ, and Σ,2 utput: Ω Ω Ω Σ, and Ω 2 Ω Σ,2 2 fr i {, 2} d 3 j {, 2} \ {i} 4 Fix Ω j. Cnstruct autmatn Ṽ w.r.t. Ω j and define functin DF. 5 Otain minimal Ω i y slving Prlem 3 w.r.t. system Ṽ and functin DF. 6 Ω i Ω i. 7 Ω [Ω, Ω 2] i {, 2} : Ω i Ω i, which implies that EΩ G i (s) EΩ G i(s) fr any s L(G). Therefre, T k T, s L(G) : δ(s), we have i {, 2} : E Ω G i (s) =, i.e., G is als decentralized distinguishale w.r.t. Ω. We are nw ready t prve the crrectness f lgrithm D-MIN-CT. Therem 2. Let Ω e the utput f lgrithm D-MIN-CT. Then Ω slves Prlem. Prf. It is clear that G is decentralized distinguishale w.r.t. Ω and T, since decentralized distinguishaility is guaranteed in each centralized cnstrained minimizatin prlem. It remains t shw that Ω is minimal; we prceed y cntradictin. Let us assume that there exists anther sensr activatin plicy Ω = [Ω, Ω 2] such that G is decentralized distinguishale w.r.t. Ω and T; and (ii) Ω Ω. The secnd cnditin means that i, j {, 2}, i j such that (iii) Ω i Ω i ; and (iv) Ω j Ω j. Suppse that i = and j = 2. Then we knw that Ω is tained y fixing gent 2 s sensr activatin plicy t e Ω Σ,2, where Ω 2 Ω 2 Ω Σ,2. y Lemma, we knw that G is decentralized distinguishale w.r.t. [Ω, Ω 2] implies that G is decentralized distinguishale w.r.t. [Ω, Ω Σ,2 ]. Hwever, since Ω Ω, this cntradicts t the fact that Ω is a slutin t Prlem 2. Similarly, suppse that i = 2 and j =. Then we knw that Ω 2 is tained y fixing gent s sensr activatin plicy t e Ω, where Ω Ω. y Lemma, we knw that G is decentralized distinguishale w.r.t. [Ω, Ω 2] implies that G is decentralized distinguishale w.r.t. [Ω, Ω 2]. Hwever, since Ω 2 Ω 2, it again cntradicts the fact that Ω 2 is a slutin t Prlem 2. Remark 3. Recall that in the synthesized minimal decentralized plicy Ω = [Ω, Ω 2], each Ω i is a pair. Therefre, t implement Ω, fr each agent i, ne can first stre the ff-line cmputed Ω i at lcal site i. T run Ω i nline, we just need t rememer the current state in Ω i and the current sensing decisin is the utput f this state. Whenever a new event is served, we just update the current state ased n the transitin functin f Ω i, mve t a new state f Ω i and update the sensing decisin t e the utput f this new state, and s frth. This is als the same way fr implementing a

10 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl Σ Σ *, + *, + Σ\ Σ\ Σ\a Σ\a Σ Σ Σ\ Σ\ Σ Σ a a *+ *+ *a+ *a+ (a) Ω ( Fig.,34. (, ),3 Decentralized, ) minimal (,3,5,7 slutins (,3,5,7, ), ) ( 2,4,6 ( 2,4,6, ) (, ) 5, ) ( 5, ) ( 2,4,7 ( 2,4,7, 2), 2) supervisr; see, e.g., [3]. a a ( 7, ) ( 7, ) ( 6,( 3) 6, 3) *+ *+ () Ω 2 Remark 4. In general, the minimal slutin t Prlem is nt unique due t the fllwing reasns. First, fr each centralized cnstraint minimizatin prlem invlved in lgrithm D- MIN-CT, the minimal slutin is nt unique in general [37]. There may exist tw incmparale centralized minimal slutins t Prlem 2 r 3. Secnd, the decentralized minimal slutin tained y lgrithm D-MIN-CT als depends n the rder f the centralized cnstraint minimizatin prlems. T implement lgrithm D-MIN-CT, we can randmly select an rder fr each agent. In general, fixing gent first and fixing gent 2 first may result in different minimal slutins. Hwever, in any case, slutin Ω returned y lgrithm D- MIN-CT is guaranteed t e minimal in the sense that ther minimal slutins must e incmparale with Ω. We illustrate lgrithm D-MIN-CT y an example. Example 5. gain, cnsider the system G in Figure 2(a) and specificatins T = {T, T 2 } defined in Example 2. Let Σ, = {, a} and Σ,2 = {, }, respectively, e the set f servale events fr gent and gent 2. Initially, we set Ω 2 = Ω Σ,2 and slve the cnstrained minimizatin prlem fr gent ; this has een slved in Example 4 and we tained Ω shwn in Figure 4(a). Next, we fix Ω fr gent and slve the cnstrained minimizatin prlem fr gent 2. Then we tain the sensr activatin plicy Ω 2 as shwn in Figure 4(). We see that Ω 2 turns all sensrs ff after is served, since nce ccurs, gent 2 will knw fr sure that the system is in state 5 r 7 and there is n need t mnitr any event. Therefre, [Ω, Ω 2] is a minimal pair f sensr activatin plicies that ensure decentralized distinguishaility. Remark 5. We cnclude this sectin y discussing the cmplexity f synthesis algrithm. Suppse that we first fix - gent 2. Initially, Ω 2 = Ω Σ,2 and its autmatn nly cntains a single state. T slve the cnstraint ptimizatin prlem when Ω 2 is fixed, first, we need t cnstruct Ṽ, which is plynmial in the size f G and Ω 2. Hwever, since an server-like cnstructed is explited, the algrithm in [37] requires expnential cmplexity w.r.t the size f the system, i.e., Ṽ, and the size f the slutin Ω is als expnential in the size f Ṽ. gain, cnstructing Ṽ when gent is fixed nly requires plynmial cmplexity w.r.t. Ω, ut synthesizing Ω 2 requires expnential cmplexity again. Therefre, the verall cmplexity is duly-expnential w.r.t. the size f G. Such a duly-expnential cmplexity arises in many synthesis prlems where tw incmparale servatins are invlved; see, e.g., [8], [6]. Remark 6. In this paper, we adpt a lgical ptimality criterin that has een widely used in the literature. One pssile future directin is t cnsider a numerical cst functin that intrduces a quantitative jective. This numerical setting is much mre challenging t deal with, in particular in the decentralized setting. Specifically, using the persn-y-persn apprach fr this setting may have the fllwing prlems. First, hw t slve the cnstrained ptimizatin prlem fr a quantitative jective may e very different frm the apprach develped in this paper. Secnd, iteratins are needed in general and the cnvergence may nt e guaranteed as the dmain f languages is infinite. Mrever, even if the persny-persn iteratin cnverges, it may nly cnverge t a lcal ptimal slutin. These questins are very interesting future directins ut are already eynd the scpe f this paper. VI. PPLICTION OF THE DECENTRLIZED STTE DISMIGUTION PROLEM In this sectin, we shw that the ntins f K- cdiagnsaility, cservaility and cprgnsaility are instances f decentralized distinguishaility. Therefre, the prpsed framewrk is applicale fr slving the dynamic sensr activatin prlems fr the purpses f decentralized fault diagnsis, decentralized cntrl and decentralized fault prgnsis.. Decentralized Fault Diagnsis In the decentralized fault diagnsis prlem, the lcal agents need t wrk as a team such that any fault e diagnsed within a unded numer f steps. Frmally, we dente y Σ F Σ u the set f fault events. We assume that Σ F is partitined int m fault types: Σ F = Σ F... Σ Fm ; we dente y Π the partitin and y F = {,..., m} the index set f the fault types. Fr any k F, we define Ψ(E Fk ) = {sf L(G) : f E Fk } t e the set f strings that end with a fault event f type k. We write E Fk s, if {s} Ψ(E Fk ). The ntin f K-cdiagnsaility was prpsed in the literature t capture whether r nt any fault can e diagnsed within K steps [7], [7]. Definitin 2. (K-Cdiagnsaility). Let K N. We say that live language L(G) is K-cdiagnsale w.r.t. Ω, Σ F and Π if ( k F)( s Ψ(Σ Fk ))( t L(G)/s : t K) (4) ( i {, 2})( w L(G))[P Ωi (w)=p Ωi (st) Σ Fk w]. T shw that K-cdiagnsaility can e frmulated as decentralized distinguishaility, fllwing similar cnstructins in [6], [36], we first refine the state space f G y defining a new autmatn G = ( Q, Σ, δ, q 0 ), where Q Q {, 0,..., K} m, q 0 = (q 0,,..., ) and the partial transitin functin δ : Q Σ Q is defined y: fr any (q, n,..., n m ) Q and σ Σ, we have δ((q, n,..., n m ), σ) = (δ(q, σ), n +,..., n m + m ) where fr each i {,..., m}, i is defined y { 0, if [ni = K] r [n i = i = σ Σ Fi ] if [0 n i < K] r [n i = σ Σ Fi ]

11 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl a f ,-,- f,-,- 2 3,-, 0 f 4, 0,- a 5,-, a 6,, - 7,-, achieving a given specificatin language. We recall its definitin frm [20]. Definitin 3. (Cservaility). We say that L(G) is cservale w.r.t. L(H), Σ c,, Σ c,2 and Ω if ( s L(H))( σ Σ c : sσ L(G) \ L(H)) (5) ( i I c (σ k ))[P Ω i (P Ωi (s)){σ} L(H) = ] Fig. 5. ugmented system G Intuitively, G simply unflds G y cunting the numer f steps since each type f fault has ccurred. Since L( G) = L(G), we can synthesize a sensr activatin plicy fr G ased n G. Fr any state q = (q, n,..., n m ) G, we dente y [ q] i its (i + ) th cmpnent, i.e., n i. ased n G, we define a set f specificatins T diag = {T, T 2,..., T m } as fllws: fr each T k T, we have Hereafter, we assume that H = (Q H, Σ, δ H, q 0,H ) is a strict su-autmatn f G, i.e., (i) Q H Q; (ii) s L(H) : δ H (s) = δ(s); and (iii) s L(G) \ L(H) : δ(s) X \ X H. This assumptin is withut lss f generality, since we can always refine H and G such that it hlds [5]. The refinement essentially takes the prduct f H f G and the resulting system cntains at mst Q H Q states. Nw, suppse that Σ c = {σ,..., σ m } is the set f cntrllale events. We define a set f specificatins T cnt = {T, T 2,..., T m } as fllws: fr each T k T, we have = {q Q : [q] k =K} and = {q Q : [q] k = } The fllwing result reveals that, t enfrce K- cdiagnsaility, it suffices t enfrce decentralized distinguishaility fr T diag. Therem 3. live language L(G) is K-cdiagnsale w.r.t. Ω, Σ F and Π if and nly if G is decentralized distinguishale w.r.t. Ω and T diag. Example 6. Let us cnsider again system G shwn in Figure 2(a). Suppse that Σ F = Σ F Σ F2 = {f } {f 2 }. Let us cnsider K =. Then the refined autmatn G is shwn in Figure 5. Fr example, state q = (6,, ) means that: (i) the system is at state 6 in G; (ii) f has ccurred fr mre than ne step (since [q] = K); and (iii) f 2 has nt ccurred (since [q] 2 = ). Then T diag = {T, T 2 } is defined y T = {(6,, )} {(,, ), (2,, ), (3,, 0), (5,, ), (7,, )} and T 2 = {(5,, ), (7,, )} {(,, ), (2,, ), (4, 0, ), (6,, )}. Since G and G are ismrphic fr this specific example, we see that T diag is indeed the same specificatin T defined in Example 2. Therefre, the slutin we tained in Example 5 has slved the sensr activatin prlem fr -cdiagnsaility.. Decentralized Supervisry Cntrl ={q Q H : δ(q, σ k )! δ H (q, σ k )!} ={q Q H : δ H (q, σ k )!} with I Tk = I c (σ k ), where! means is nt defined. Intuitively, fr each cntrllale event σ k Σ c, is the set f states at which σ k must e disaled fr safety purpses, while is the set f states at which σ k must e enaled t achieve L(H). The fllwing result reveals that cservaility is als a special case f decentralized distinguishaility with T cnt. Therem 4. Let G e the system and H e the specificatin autmatn. Then L(G) is cservale w.r.t. L(H), Σ c,, Σ c,2 and Ω if and nly if G is decentralized distinguishale w.r.t. Ω and T cnt. C. Decentralized Fault Predictin In sme safety-critical systems, we may nt nly want t diagnse any fault after its ccurrence, ut als want t predict any fault efre it ccurs [9]. In [5], the ntin f cprgnsaility was prpsed t capture whether r nt any fault ccurrence can e predicted in a decentralized system. The definitin is reviewed as fllws. Definitin 4. (Cprgnsaility). We say that language L(G) is cprgnsale w.r.t. Ω and Σ F if nther imprtant decentralized decisin-making prlem is the decentralized supervisry cntrl prlem [20], [4]. In this prlem, each lcal agent i I can disale events in Σ c,i Σ dynamically ased n its lcal servatin Ω i. We define Σ c = i I Σ c,i as the set f all cntrllale events and fr each σ Σ c, we define I c (σ) = {i I : σ Σ c,i } as the set f agents that can disale σ. The cntrl jective is t make sure that the clsed-lp system achieves a desired language L(H) L(G). The key prperty regarding the decentralized infrmatin in this prlem is the ntin f cservaility; it tgether with the ntin f cntrllaility prvide the necessary and sufficient cnditins fr exactly ( s Ψ(Σ F ))( t {s} : Σ F t) (6) ( i {, 2})( u P Ω i (P Ωi (t)) : Σ F / u) ( K N)( v L(G)/u)[ v K Σ F uv] T prceed further, we assume that the state space f G is partitined as Q = Q N Q F such that s L(G) : δ(s) Q N Σ F s; and s L(G) : δ(s) Q F Σ F s. Nte that, this assumptin is withut lss f generality since we can simply refine the state space f G such that this assumptin hlds.

12 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl In rder t frmulate cprgnsaility as an instance f decentralized distinguishaility, we need the ntins f nnindicatr states and undary states, which are initially intrduced in [5]. We say that a state q Q is a nn-indicatr state, if q Q N and K N, s L(G, q) : s K Σ F / s; and a undary state, if f Σ F : δ(q, f)!. We dente y N Q and Q the set f nn-indicatr states and the set f undary states, respectively. With these ntins, we define a simple specificatin T pre := {T }, where Q T = Q and Q T = N Q with I T = I. The fllwing result reveals that, t enfrce cprgnsaility, it suffices t enfrce decentralized distinguishaility with T pre. Therem 5. L(G) is cprgnsale w.r.t. Ω and ΣF if and nly if G is decentralized distinguishale w.r.t. Ω and T pre. Remark 7. Nte that Q and N Q need nt e disjint. y the ave therem, the system will nt e cprgnsale under any sensr activatin plicies if Q N Q. D. Extensin t the Cnjunctive rchitecture S far, we have shwn that K-cdiagnsaility, cservaility and cprgnsaility are special cases f decentralized distinguishaility. s we mentined earlier, all results in this paper are develped ased n the disjunctive architecture, i.e., the crdinatr issues glally if and nly if ne lcal agent issues. lternatively, ne may als use the cnjunctive rule t tain a glal decisin, i.e., the crdinatr issues 0 glally if and nly if ne lcal agent issues 0. In this case, suppse that a string leading t a state in is executed and a glal decisin 0 has t e made. Then, a lcal agent must knw that the system is nt in unamiguusly when it issues 0 ; therwise a wrng glal decisin may e made at sme state in. Therefre, we need t require that ( s L(G) : δ(s) )( i I T k )[E G Ω i (s) = ] y cmparing the ave requirement with decentralized distinguishaility, which is defined in terms f the disjunctive architecture, we see that this requirement is indeed the same as decentralized distinguishaility y swapping and QT k. Therefre, there is n need t define a cnjunctive versin f decentralized distinguishaility; it is just a matter f hw the specificatin T k is defined. Fr example, in [4], the ntin f D&-cservaility was prpsed as a cmplement f cservaility. We recall its definitin. Definitin 5. (D&-Cservaility). We say that L(G) is D&-cservale w.r.t. L(H), Σ c,, Σ c,2 and Ω if ( s L(H))( σ Σ c : sσ L(H)) (7) ( i I c (σ k ))[P Ω i (P Ωi (s)){σ} L(G) L(H)] Here, D& stands fr Disjunctive & nti-permissive. ls, cservaility in Definitin 3 is referred t as C&P-cservaility, where C& standards fr Cnjunctive & Permissive. The reasn why C&Pcservaility crrespnds t decentralized distinguishaility in the disjunctive architecture is that, [20] cnsiders the cnjunctin f enalements, while T cnt captures the disjunctin f disalements; they are essentially equivalent. Intuitively, D&-cservaility requires that fr any string fr which σ has t e enaled, there exists at least ne agent that knws fr sure that σ shuld nt e disaled. We can als frmulate D&-cservaility as an instance f decentralized distinguishaility y defining T CJ cnt = {T CJ, T2 CJ,..., T CJ where fr each Tk CJ T, we have Q T CJ k ={q Q H : δ H (q, σ k )!} Q T CJ k ={q Q H : δ(q, σ k )! δ H (q, σ k )!} m }, with I Tk = I c (σ k ). The prf f the crrectness f T CJ cnt is mitted since it is similar t the prf f Therem 4. Similarly, ne can als shw that cnjunctive K- cdiagnsaility [34], [35] and cnjunctive cprgnsaility [4], [39] are instances f decentralized distinguishaiity; we just need t define new specificatins T CJ diag swapping each and QT k and TCJ pre y in T diag and T pre, respectively. VII. CONCLUSION We presented a nvel apprach fr slving the prlem f decentralized sensr activatin fr a class f prperties. We prpsed the ntin f decentralized distinguishaility, which cvers cservaility, K-cdiagnsaility and cprgnsaility. T enfrce decentralized distinguishaility, we first adpted a persn-y-persn apprach t decmpse the decentralized minimizatin prlem t tw cnsecutive centralized cnstrained minimizatin prlems. Then, a nvel apprach was prpsed t reduce each centralized cnstrained minimizatin prlem t a fully centralized sensr activatin that is slved effectively in the literature. Finally, we shwed that the decentralized slutin tained y ur methdlgy is language-ased minimal. PPENDIX. Prfs nt cntained in main dy Prf f Therem 3 Prf. ( ) y cntrapsitin. Suppse that G is nt decentralized distinguishale. Then we knw that there exist k {,..., m} and a string s L(G) such that q := δ(s) and fr each i {, 2}, there exists q i E G Ωi (s) such that q i. Then we knw that, fr each i {, 2}, there exists a string s i L(G) such that δ(s i ) = q i and P Ωi (s) = P Ωi (s i ). y the definitin f T k, q implies that [q] k = K. ccrding t the cnstructin f G, δ(s) = q implies that we can write s = uv such that u Ψ(Σ Fk ) and v K. Fr each i {, 2}, since q i, we knw that [q i] k =, which implies that Σ Fk / s i. Overall, we knw that ( k F)( u Ψ(Σ Fk ))( v L(G)/u : v K) ( i {, 2})( s i L(G))[P Ωi (uv)=p Ωi (s i ) Σ Fi s i ] (8) i.e., L(G) is nt K-cdiagnsale. ( ) Still y cntrapsitin. Suppse that G is nt K- cdiagnsale, i.e., Equatin (8) hlds. Let q := δ(uv), q := δ(s ) and q 2 := δ(s 2 ). Then, accrding t the definitin f G, we knw that [q]k = K, [q ] k = [q 2 ] k =, which implies that q and q, q 2. Mrever,

13 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl since fr each i =, 2, P Ωi (uv) = P Ωi (s i ), we knw that q i E G Ωi (s i ) = E G Ωi (uv), i.e., E G Ωi (uv). Overall, we knw that ( T k T)( uv L( G) : δ(uv) )( i {, 2})[E G Ωi (uv) ], i.e., G is nt decentralized distinguishale w.r.t. T diag. Prf f Therem 4 Prf. ( ) y cntrapsitin. Suppse that G is nt decentralized distinguishale. Then we knw that there exist T k T, s L(G) : δ(s) such that fr each i I T k, there exists t i L(G) such that P Ωi (s) = P Ωi (t i ) and δ(t i ). Let σ k Σ c e the cntrllale event assciated t T k. Then δ(s) implies that s L(H), sσ k L(G) \ L(H) and δ(t i ) implies that t iσ k L(H). Mrever, I Tk = I c (σ k ). Overall, we knw that s L(H), σ k Σ c such that sσ k L(G) \ L(H) and fr each i I c (σ k ), t i σ k P Ω i (P Ωi (s)){σ k } L(H), i.e., L(G) is nt cservale. ( ) y cntrapsitin. Suppse that L(G) is nt cservale. Then we knw that s L(H), σ k Σ c : sσ k L(G) \ L(H) such that fr each i I c (σ k ), there exists t i L(G) such that t i σ k L(H) and P Ωi (s) = P Ωi (t i ). Fr the ave s and t i, we knw that δ(s) and δ(t i ). Therefre, fr s and σ k, we knw that fr each i I Tk = I c (σ k ), δ(t i ) EΩ G i (t i ) = E Ω G i (s), i.e., G is nt decentralized distinguishale. Prf f Therem 5 Prf. ( ) y cntrapsitin. Suppse that G is nt decentralized distinguishale. Then we knw that there exists s L(G) such that q := δ(s) Q and fr each i {, 2}, there exists q i EΩ G i (s) such that q i N Q, i.e., there exists a string s i L(G) such that Σ F / s i, δ(s i ) = q i and P Ωi (s) = P Ωi (s i ). Since q Q, we knw that f Σ F : sf Ψ(Σ F ). Let t {s} e an aritrary prefix f s such that Σ F / t. Then fr each i {, 2}, since P Ωi (s) = P Ωi (s i ), we knw that t {s}, t i {s i } : P Ωi (t) = P Ωi (t i ) Σ F / t i (9) Mrever, since q i N Q which is reachale frm δ(t i ), we knw that, fr any K N, there exists a string w i such that t i w i L(G), Σ F / t i w i and w i K. Overall, we knw that ( sf Ψ(Σ F ))( t {sf} : Σ F t) (20) ( i {, 2})( t i P Ω i (P Ωi (t)) : Σ F / t i ) ( K N)( w i L(G)/t i )[ w i K Σ F t i w i ] i.e., G is nt cprgnsale w.r.t. Ω and Σ F. ( ) Suppse that G is nt cprgnsale, i.e., Equatin (20) hlds. Let sf e a string satisfying Equatin (20). Let t e a prefix f s such that Σ F / t and tf {s} fr sme f Σ F. Then we knw that q := δ(t) Q. ccrding t Equatin (20), we knw that, fr each agent i {, 2}, there exists a string t i L(G) such that ) Σ F / t i ; and 2) ( K N)( w i L(G)/t i )[ w i K Σ F t i w i ]; and 3) P Ωi (t i ) = P Ωi (t). The first tw cnditins imply that q i := δ(t i ) N Q. Mrever, the last cnditin implies that {q, q i } EΩ G i (t). Overall, we knw that ( t L(G) : δ(t) Q )( i {, 2})[E G Ω i (t) N Q = ], i.e., G is nt decentralized distinguishale. REFERENCES [] J.C. asili, S.T.S. Lima, S. Lafrtune, and M.V. Mreira. Cmputatin f minimal event ases that ensure diagnsaility. Discrete Event Dyn. Sys.: Thery & ppl., 22(3): , 202. [2] R. el and J. H. van Schuppen. Decentralized failure diagnsis fr discrete-event systems with cstly cmmunicatin etween diagnsers. In 6th Int.l Wrkshp n Discrete Event Systems, pages 75 8, [3] C. Cassandras and S. Lafrtune. Intrductin t Discrete Event Systems. Springer, 2nd editin, [4] F. Cassez and S. Tripakis. Fault diagnsis with static and dynamic servers. Fundam. Infrma., 88(4): , [5] H. Ch and S.I. Marcus. On supremal languages f classes f sulanguages that arise in supervisr synthesis prlems with partial servatin. Math. Cntr. Sig. Syst., 2():47 69, 989. [6] E. Dallal and S. Lafrtune. On mst permissive servers in dynamic sensr activatin prlems. IEEE Trans. utm. Cntr., 59(4):966 98, 204. [7] R. Deuk, S. Lafrtune, and D. Teneketzis. Crdinated decentralized prtcls fr failure diagnsis f discrete event systems. Discrete Event Dyn. Sys.: Thery & ppl., 0(-2):33 86, [8] J. Dureil, P. Darndeau, and H. Marchand. Supervisry cntrl fr pacity. IEEE Trans. utm. Cntr., 55(5):089 00, 200. [9] S. Genc and S. Lafrtune. Predictaility f event ccurrences in partially-served discrete-event systems. utmatica, 45(2):30 3, [0]. Haji-Valizadeh and K. Lpar. Minimizing the cardinality f an events set fr supervisrs f discrete-event dynamical systems. IEEE Trans. utm. Cntr., 4(): , 996. [] Y.-C. H. Team decisin thery and infrmatin structures. Prceedings f the IEEE, 68(6): , 980. [2] Y. Huang, K. Rudie, and F. Lin. Decentralized cntrl f discrete-event systems when supervisrs serve particular event ccurrences. IEEE Trans. utm. Cntr., 53(): , [3] S. Jiang, R. Kumar, and H. Garcia. Optimal sensr selectin fr discreteevent systems with partial servatin. IEEE Trans. utm. Cntr., 48(3):369 38, [4]. Khumsi and H. Chaki. Cnjunctive and disjunctive architectures fr decentralized prgnsis f failures in discrete-event systems. IEEE Trans. utm. Sci. Eng., 9(2):42 47, 202. [5] R. Kumar and S. Takai. Decentralized prgnsis f failures in discrete event systems. IEEE Trans. utm. Cntr., 55():48 59, 200. [6]. Overkamp and J. H. van Schuppen. Maximal slutins in decentralized supervisry cntrl. SIM J. Cntr. Optim., 39(2):492 5, [7] W. Qiu and R. Kumar. Decentralized failure diagnsis f discrete event systems. IEEE Trans. S.M.C.-Part : Syst. Hum., 36(2): , [8] K. Rudie, S. Lafrtune, and F. Lin. Minimal cmmunicatin in a distriuted discrete-event system. IEEE Trans. utm. Cntr., 48(6): , [9] K. Rudie and J.C. Willems. The cmputatinal cmplexity f decentralized discrete-event cntrl prlems. IEEE Transactins n utmatic Cntrl, 40(7):33 39, 995. [20] K. Rudie and W.M. Wnham. Think glally, act lcally: Decentralized supervisry cntrl. IEEE Trans. utm. Cntr., 37(): , 992. [2] W. Sadid, S.L. Ricker, and S. Hashtrudi-Zad. Nash equilirium fr cmmunicatin prtcls in decentralized discrete-event systems. In merican Cntrl Cnference, pages IEEE, 200. [22] D. Sears and K. Rudie. Efficient cmputatin f sensr activatin decisins in discrete-event systems. In 52nd IEEE Cnf. Decisin and Cntrl, pages , 203. [23] D. Sears and K. Rudie. On cmputing indistinguishale states f nndeterministic finite autmata with partially servale transitins. In 53rd IEEE Cnf. Decisin and Cntrl, pages , 204. [24] D. Sears and K. Rudie. Minimal sensr activatin and minimal cmmunicatin in discrete-event systems. Discrete Event Dyn. Sys.: Thery & ppl., 26(2): , 206. [25] S. Shu, Z. Huang, and F. Lin. Online sensr activatin fr detectaility f discrete event systems. IEEE Trans. utm. Sci. Eng., 0(2):457 46, 203. [26] J.G. Thistle. Undecidaility in decentralized supervisin. Systems & Cntrl Letters, 54(5): , 2005.

14 This article has een accepted fr pulicatin in a future issue f this jurnal, ut has nt een fully edited. Cntent may change prir t final pulicatin. Citatin infrmatin: DOI 0.09/TC , IEEE Transactins n utmatic Cntrl [27] D. Thrsley and D. Teneketzis. ctive acquisitin f infrmatin fr diagnsis and supervisry cntrl f discrete event systems. Discrete Event Dyn. Sys. : Thery & ppl., 7(4):53 583, [28] S. Tripakis. Undecidale prlems f decentralized servatin and cntrl n regular languages. Inf. Prc. Letters, 90():2 28, [29] J.N. Tsitsiklis. On the cntrl f discrete-event dynamical systems. Mathematics f Cntrl, Signals and Systems, 2(2):95 07, 989. [30] J. H. van Schuppen. Cntrl f distriuted stchastic systemsintrductin, prlems, and appraches. In 8th IFC Wrld Cngress, pages , 20. [3] W. Wang, S. Lafrtune,. R. Girard, and F. Lin. Optimal sensr activatin fr diagnsing discrete event systems. utmatica, 46(7):65 75, 200. [32] W. Wang, S. Lafrtune, and F. Lin. n algrithm fr calculating indistinguishale states and clusters in finite-state autmata with partially servale transitins. Systems & Cntrl Letters, 56(9):656 66, [33] W. Wang, S. Lafrtune, F. Lin, and. R. Girard. Minimizatin f dynamic sensr activatin in discrete event systems fr the purpse f cntrl. IEEE Trans. utm. Cntr., 55(): , 200. [34] Y. Wang, T.-S. Y, and S. Lafrtune. Diagnsis f discrete event systems using decentralized architectures. Discrete Event Dynamic Systems: Thery & pllicatins, 7(2): , [35] T. Yamamt and S. Takai. Cnjunctive decentralized diagnsis f discrete event systems. In 4th IFC Wrkshp n Dependale Cntrl f Discrete Systems, vlume 4, pages 67 72, 203. [36] X. Yin and S. Lafrtune. Cdiagnsaility and cservaility under dynamic servatins: Transfrmatin and verificatin. utmatica, 6:24 252, 205. [37] X. Yin and S. Lafrtune. general apprach fr slving dynamic sensr activatin prlems fr a class f prperties. In 54th IEEE Cnf. Decisin and Cntrl, pages , 205. [38] X. Yin and S. Lafrtune. Minimizatin f sensr activatin in decentralized fault diagnsis f discrete event systems. In 54th IEEE Cnf. Decisin and Cntrl, pages 04 09, 205. [39] X. Yin and Z. Li. Decentralized fault prgnsis f discrete event systems with guaranteed perfrmance und. utmatica, 69: , 206. [40] S. Ykta, T. Yamamt, and S. Takai. Cmputatin f the delay unds and synthesis f diagnsers fr decentralized diagnsis with cnditinal decisins. Discrete Event Dyn. Sys.: Thery & ppl., 27():45 84, 207. [4] T.-S. Y and S. Lafrtune. general architecture fr decentralized supervisry cntrl f discrete-event systems. Discrete Event Dyn. Sys.: Thery & ppl., 2(3): , [42] T.-S. Y and S. Lafrtune. NP-cmpleteness f sensr selectin prlems arising in partially served discrete-event systems. IEEE Trans. utm. Cntr., 47(9): , [43] T.-S. Y and S. Lafrtune. Decentralized supervisry cntrl with cnditinal decisins: Supervisr realizatin. IEEE Transactins n utmatic Cntrl, 50(8):205 2, Stéphane Lafrtune received the.eng degree frm Ecle Plytechnique de Mntréal in 980, the M.Eng degree frm McGill University in 982, and the Ph.D degree frm the University f Califrnia at erkeley in 986, all in electrical engineering. Since Septemer 986, he has een with the University f Michigan, nn rr, where he is a Prfessr f Electrical Engineering and Cmputer Science. Dr. Lafrtune is a Fellw f the IEEE (999). He received the Presidential Yung Investigatr ward frm the Natinal Science Fundatin in 990 and the Gerge S. xely Outstanding Paper ward frm the Cntrl Systems Sciety f the IEEE in 994 (fr a paper c-authred with S.-L. Chung and F. Lin) and in 200 (fr a paper c-authred with G. arrett). Dr. Lafrtune s research interests are in discrete event systems and include multiple prlem dmains: mdeling, diagnsis, cntrl, ptimizatin, and applicatins t cmputer and sftware systems. He is the lead develper f the sftware package UMDES and c-develper f DESUM with L. Ricker. He c-authred, with C. Cassandras, the textk Intrductin t Discrete Event Systems - Secnd Editin (Springer, 2008). Dr. Lafrtune is Editr-in- Chief f the JOURNL OF DISCRETE EVENT DYNMIC SYSTEMS: THEORY ND PPLICTIONS. Xiang Yin was rn in nhui, China, in 99. He received the.eng degree frm Zhejiang U- niversity in 202, the M.S. degree frm the University f Michigan, nn rr, in 203, and the Ph.D degree frm the University f Michigan, nn rr, in 207, all in electrical engineering. Since 207, he has een with the Shanghai Jia Tng University, where he is an ssciate Prfessr in the Department f utmatin. His research interests include supervisry cntrl f discrete-event systems, mdel-ased fault diagnsis, frmal methds, security and their applicatins t cyer and cyer-physical systems. Dr. Yin received the Outstanding Reviewer ward frm UTOMTIC in 206, the Outstanding Reviewer ward frm IEEE TRNSCTIONS ON UTOMTIC CONTROL in 207 and the IEEE Cnference n Decisin and Cntrl (CDC) est Student Paper ward Finalist in 206. He is the c-chair f the IEEE CSS Technical Cmmittee n Discrete Event Systems.