On Decentralized Observability of Discrete Event Systems

2011 50th IEEE Conference on Decision nd Control nd Europen Control Conference (CDC-ECC) Orlndo, FL, USA, Decemer 12-15, 2011 On Decentrlized Oservility of Discrete Event Systems M.P. Csino, A. Giu, C. Mhule, C. Setzu Astrct In this pper we del with the prolem of decentrlized oservility of discrete event systems. We consider set of sites tht oserve suset of events. Ech site trnsmits its own oservtion to coordintor tht decides if the word oserved elongs to legl ehvior or not. We study two different properties: uniform q oservility nd q dignosility. Then, we prove tht oth properties re decidle for regulr lnguges. Finlly, we give n lgorithm to compute strting from given initil stte, the time instnts t which the synchroniztion hs to e done so s to gurntee tht if n illegl word hs occurred it is immeditely detected. A. Motivtion I. INTRODUCTION In [1] Tripkis defines property tht he clls locl oservility. The ide is the following: set of n locl sites oserve, through their own projection msks P i, word w of symols tht is known to elong to lnguge L. A lnguge K L is loclly oservle if, ssuming ll locl sites send to coordintor ll oserved strings P i (w), the coordintor cn decide for ny w if the word elongs to K or to L \ K. Note tht this property ws shown in [1] to e undecidle even when lnguges L nd K re regulr: this is due to the fct tht the length of word w cn e ritrrily long. On the contrry, ssuming only words of ounded length re considered, the property is decidle for ritrry lnguges, since it must only e checked over finite numer of strings. We oserve tht this property is closely relted to locl dignosility s defined y Smpth et l. [2]. In fct, lnguge K in this setting represents the set of ll fult-free evolutions, while the lrger set L lso includes the fulty ones. The prolem we wnt to ddress is the following. Assume w descries the event driven evolution of system. The coordintor cn t ny moment send request to ll locl sites to know the oserved words since the previous request: such mechnism is clled synchroniztion. After ech synchroniztion coordintor should e le to decide if, on the sis of the informtion received so fr from the locl sites, the word w generted is legl, i.e, elongs to K. Note This work hs een prtilly supported y the Europen Community s Seventh Frmework Progrmme under project DISC (Grnt Agreement n. INFSO-ICT-224498). At University of Zrgoz it ws prtilly supported lso y CICYT - FEDER projects DPI2010-20413 nd y Fundción Argón I+D. M.P. Csino, A. Giu nd C. Setzu re with the Deprtment of Electricl nd Electronic Engineering, University of Cgliri, Pizz D Armi, 09123 Cgliri, Itly {csino,giu,setzu@diee.unic.it}. C. Mhule is with the Argón Institute of Engineering Reserch (I3A), University of Zrgoz, Mri de Lun 1, 50018 Zrgoz, Spin {cmhule@unizr.es}. tht synchroniztion is costly, thus lthough we ssume tht the mximl numer of events tht cn e generted y the system etween two consecutive synchroniztions is ounded, the coordintor should request s few synchroniztions s needed to solve the oservility prolem. Also the distnce etween two consecutive synchroniztions, expressed in terms of the numer of events generted etween them, needs not e constnt ut my opportunisticlly vry with the word generted so fr. In this setting, lthough the sic notion of locl oservility given y Tripkis is still fundmentl, two mjor extensions re needed. In fct the oservility property defined in [1] mkes two rther restrictive ssumptions. The first ssumption is tht the oservility property is defined only with respect to words in L. On the contrry, in our setting synchroniztion occurs repetedly. Thus if synchroniztion occurs fter word w hs een generted we re interested in the oservility of the residul lnguge w 1 K, i.e., the set of ll strings tht elong to K nd whose prefix is w, with respect to the residul lnguge w 1 L. Correspondingly, we introduce the notion of uniform q oservility. The second ssumption in [1] is tht when the oservtion strts the word generted so fr (tht s discussed in the previous prgrph is lwys the empty word) is perfectly known. On the contrry, in our setting when synchroniztion occurs the coordintor should e le to determine if the generted string is legl or not, ut my not e le to unmiguously estimte it. Thus when next oservtion strts the word generted so fr is only known to elong to given set. To cpture this condition, we introduce the notion of q dignosility. B. Literture review Oservility is fundmentl property tht hs received lot of ttention during the lst decdes due to the importnce of reconstructing plnt sttes tht cnnot e mesured. Severl contriutions hve een presented in the frmework of utomt [3], [4], [5], [6]. In [3] Cines et l. showed how it is possile to use the informtion contined in the pst sequence of oservtions (given s sequence of oservtion sttes nd control inputs) to compute the set of consistent sttes, while in [4] the oserver output is used to steer the stte of the plnt to desired terminl stte. A similr pproch ws lso used y Kumr et l. [6] when defining oserver sed dynmic controllers in the frmework of supervisory predicte control prolems. Özveren nd Willsky [5] proposed n pproch for uilding oservers tht llows one to reconstruct the stte of finite 978-1-61284-799-3/11/$26.00 2011 IEEE 378

utomt fter word of ounded length hs een oserved, showing tht n oserver my hve n exponentil numer of sttes. A prolem strictly relted to oservility s defined in the present pper is opcity. A system is (current-stte) opque if its (current) stte is never exposed to certinly elong to given set of secret sttes. See the work of Soori nd Hdjicostis [7], [8] nd of Dureil et l. [9]. Finlly, very generl pproch for oservility with communiction hs een presented y Brret nd Lfortune in [10] in the context of supervisory control, nd severl techniques for designing possily optiml communiction policy hve lso een discussed therein. By optiml we men tht the locl sites communicte s lte s possile, only when strictly necessry to prevent the undesirle ehvior. Our work is y lrge specil cse of the rchitecture in [10] ecuse we llow communictions only etween the coordintor nd the locl oservers nd not mong locl oservers nd we do not consider control prolem ut simply n oservtion one. There re, however, few differences in our pproch derived from [1] with respect to [10] tht motivte the need for dditionl investigtion. These differences re listed here. First, we frme our results in the context of lnguges, rther thn utomt: this mens tht some of our definitions nd results pply to possily non regulr lnguges. Secondly, while in [10] communictions re decided y the locl oservers nd re triggered y the oservtion of n event, in our cse the communictions re triggered y the coordintor. Finlly, we ssume tht the coordintor knows the numer of events generted so fr, ut cnnot directly oserve their lel; thus the oservtion structure of the coordintor is not projection msk ut simply function f : L N tht counts the events generted so fr. Recently Ricker nd Cillud [11] hve lso considered setting where communictions my lso e triggered y the receiver, tht requests informtion from sender. Furthermore, they lso discuss policies where communiction occurs fter prefixes of ny of the ehviors involved in violtion of co-oservility, not just those tht my result in undesired ehvior. II. BASIC NOTATIONS Let Σ e finite lphet: Σ denotes the set of ll finite strings over Σ, i.e., the Kleene str, nd ε denotes the empty string. Given two strings u nd v, uv is the conctention of u nd v. A deterministic finite utomton (DFA) is tuple G = (X,Σ,δ,x 0,X m ) where X is the set of sttes, Σ is the finite set of events, prtil function δ : X Σ X is the trnsition function, x 0 X is the initil stte, nd X m X is the set of mrked sttes. The generted nd mrked lnguges of G, denoted y L(G) nd L m (G), respectively, re defined s L(G) = {w Σ δ(x 0,w) is defined} nd L m (G) = {w Σ δ(x 0,w) X m }. Given two deterministic finite utomt G 1 = (X 1,Σ 1,δ 1,x 0,1,X m,1 ) nd G 2 = (X 2,Σ 2,δ 2,x 0,2,X m,2 ), the prllel composition of G 1 nd G 2 is the utomton G 1 G 2 = (X,Σ 1 Σ 2,δ,(x 0,1,x 0,2 ),X m), where X (X 1 X 2 ), X m (X m,1 X m,2 ) nd ( x 1,x 2 ) if e Σ 1 \ Σ 2,δ 1 (x 1,e) = x 1 ; (x 1, x 2 ) if e Σ 2 \ Σ 1,δ 2 (x 2,e) = x 2 ; δ (x,e) = ( x 1, x 2 ) if e Σ 1 Σ 2,δ 1 (x 1,e) = x 1, δ 2 (x 2,e) = x 2 ; not defined otherwise. Given word w Σ, nd n lphet Σ i Σ, we denote s P i (w) the projection of w over Σ i, tht cn e recursively defined s follows. If w = ue, where u Σ nd e Σ, it holds { Pi (u)e if e Σ P i (w) = i, P i (u) otherwise Given lnguge L nd string w Σ, the residul of L with respect to (wrt) w is the lnguge w 1 L = {z wz L}. The lnguge L is regulr iff the set of its residuls s w rnges over Σ is finite, i.e., iff the set {w 1 L w Σ } is finite. The crdinlity of the set {w 1 L w Σ } is clled the index of L. III. UNIFORM q OBSERVABILITY Let us consider two prefix-closed lnguges K nd L defined over n lphet Σ, such tht K L Σ, nd set of n su-lphets Σ i Σ, i = 1,...n. The n su-lphets Σ i s re ssocited to n sites S i, i = 1,...,n. In prticulr, Σ i includes ll the events tht cn e oserved y S i. A first definition of decentrlized oservility hs een given y Tripkis in [1] in the cse of regulr lnguges. Definition 3.1: Let us consider two regulr lnguges L nd K. The lnguge K is jointly oservle wrt L nd Σ i, for i = 1,...,n, if there exists totl function f : Σ 1...Σ n {0,1}, such tht w L w K f(p 1 (w),...,p n (w)) = 1. (1) The ove property uses unounded memory since the word w my hve ritrry length, thus it is undecidle [1]. In this pper we generlize such definition to the cse of finite memory, i.e., the coordintor cn t ny moment send request to ll locl sites to know the oserved words since the previous request. On the sis of the informtion received so fr from the locl sites, the coordintor should estlish if the evolution is legl. Definition 3.2: Let Σ e finite lphet, nd Σ i Σ, with i = 1,...,n, e n su-lphets of Σ. Let L nd K e two prefix closed lnguges such tht K L Σ. The lnguge K is clled uniformly q oservle wrt L nd Σ i, for i = 1,...,n, if w K there exists function f w : Σ 1... Σ n {0,1} such tht u w 1 L with u q, it holds u w 1 K f w (P 1 (u),...,p n (u)) = 1. (2) 379

x 1 x 0 x 2 Fig. 1. The DFA considered in Exmple 3.5., In simple words, uniform q oservility implies the possiility of estlishing if the ehvior of given system is legl, only looking t the occurrence of no more thn q events, nd knowing tht the sequence w preceding such events is legl. Let us now introduce n equivlence reltion mong strings tht llows us to rephrse the ove definition of uniform oservility. Definition 3.3: Let Σ e finite lphet, nd Σ i Σ, with i = 1,...,n, e n su-lphets of Σ. Let L nd K e two prefix closed lnguges such tht K L Σ. A word u w 1 L is oservtion equivlent (or simply equivlent) to v w 1 L, i.e., u v, if P i (u) = P i (v) for ll i = 1,...,n. We denote [u] the set of words tht re equivlent to u. Finlly, we sy tht two words tht re not equivlent re distinguishle. Using this notion, the definition of uniform q oservility of lnguge cn e rewritten s follows. Definition 3.4: Let Σ e finite lphet, nd Σ i Σ, with i = 1,...,n, e n su-lphets of Σ. Let L nd K e two prefix closed lnguges such tht K L Σ. The lnguge K is clled uniform q oservle wrt L nd Σ i, i = 1,...,n, if w K, nd u w 1 L, w[u] K w[u] K. (3) The following exmple clrifies the ove definitions. Exmple 3.5: Let Σ = {,}, Σ 1 = {}, Σ 2 = {}, L e the lnguge generted y the regulr expression ( + ), while K is the lnguge generted y the regulr expression K 1 + K 2 where K 1 = ( ) nd K 2 = ( ). It cn e esily verified tht L corresponds to the lnguge generted y the DFA in Fig. 1 strting from x 0, while K is the lnguge generted y the sme DFA neglecting the stte x 2, still ssuming x 0 s the initil stte. Moreover, K 1 corresponds to the set of words tht finish in x 0, while K 2 corresponds to the set of words tht finish in x 1. We wnt to study the uniform q oservility of K wrt L, Σ 1 nd Σ 2. Let s strt with q = 1. According to the definition of uniform q oservility we hve to consider ll possile words u w 1 L of unitry length. This is equivlent to consider n ritrry word w (K 1 + K 2 ) followed y ny word u of length 1. Since ll words w K 1 terminte in x 0, then only two words of unitry length my occur fter w, nmely u 1 = nd u 2 =. Clerly it is wu 1 K nd wu 2 L \ K, therefore it should e f w (P 1 (u 1 ),P 2 (u 1 )) = f w (,ε) = 1 nd f w (P 1 (u 2 ),P 2 (u 2 )) = f w (ε,) = 0. Let us now consider n ritrry word w K 2, i.e., n ritrry word tht termintes in x 1. Strting from x 1 the only dmissile words of length 1 re u 3 = nd u 4 =. In such cse oth wu 3 nd wu 4 re in K, thus it should e nd f w (P 1 (u 3 ),P 2 (u 3 )) = f w (,ε) = 1 f w (P 1 (u 4 ),P 2 (u 4 )) = f w (ε,) = 1. This enles us to conclude tht K is uniformly 1 oservle wrt L, Σ 1 nd Σ 2. Note tht the sme conclusion cn e drwn using the notion of uniform 1-oservility sed on equivlence clsses. Indeed, oth u 1 nd u 2, nd u 3 nd u 4 re distinguishle. Let us now study uniform 2 oservility. As discussed ove, if w K 1 we should consider ll words u L of length 2 tht cn e generted from x 0, i.e., u {,,,}. However, nd re clerly equivlent ut w K while w L \ K. Thus K is not uniformly 2 oservle wrt L, Σ 1 nd Σ 2. In other terms, we cn sy tht function f w stisfying the if nd only if condition in (2) could not e defined. Indeed it should simultneously e nd f w (P 1 (),P 2 ()) = f w (,) = 1 f w (P 1 (),P 2 ()) = f w (,) = 0, i.e., f w should ssume different vlues in correspondence to the sme rguments. The following result trivilly follows from Definition 3.2. Proposition 3.6: If K is uniformly q oservle wrt L nd set of lphets Σ i, i = 1,...,n, then it is lso uniformly (q 1)-oservle wrt them. Proof: Follows y the fct tht the sme f w function used in the cse of uniform q oservility cn e used in the cse of uniform (q 1) oservility, simply restricting its rguments to words of length q 1 rther thn q. This implies tht, if lnguge is uniformly q oservle for some finite q > 1, then it is lso uniformly 1-oservle. A simple condition under which uniform 1-oservility is gurnteed is now given. Proposition 3.7: Let us consider set of lphets Σ i, i = 1,...,n, such tht Σ 1... Σ n = Σ. Any lnguge K L Σ is uniformly 1 oservle wrt to L nd Σ i, i = 1,...,n. Proof: Since Σ 1... Σ n = Σ, there exists t lest one site tht cn detect ny event e tht hs occurred. If the function f w hs een defined for word w, the new function simply ssigns the vlue 1 if we K nd 0 otherwise. Being possile to define the function for ny oserved event, the system is uniformly 1 oservle. 380

On the contrry, uniform 1-oservility is no more ensured if one or more events in Σ re not oservle y ll the sites. Let n ˆΣ = Σ \ Σ i (4) i=1 denotes the set of events tht re oservle y no site. If ˆΣ then K L Σ cn e not uniformly 1-oservle wrt L nd Σ i s, even if ll words formed y the conctention of word in K nd word in ˆΣ re still in K, i.e., K ˆΣ L K. A. Regulr lnguges Prticulrly interesting results cn e proved if K nd L re prefix-closed regulr lnguges. First, it cn e shown tht nlyzing uniform q-oservility is decidle prolem. Then, simple criterion cn e given to estlish if certin sequence is legl, sed on DFA. Proposition 3.8: Let us consider set of lphets Σ i, i = 1,...,n, such tht Σ 1... Σ n = Σ. Let K nd L e two prefix-closed lnguges such tht K L Σ. If K nd L re regulr lnguges, the uniform q oservility of K wrt L nd Σ i is decidle for ny finite q N. Proof: According to the Myhill-Nerode Theorem [12], ech regulr lnguge L hs finite index, i.e., the set of lnguges {w 1 L w L} is finite. This implies tht it is sufficient to check the existence of function f w for finite numer of words w over finite suset of Σ 1 Σ n, i.e., the set of projections on Σ i s, i = 1,...,n, with length less thn or equl to q. Thus the prolem is decidle. From the Myhill-Nerode Theorem [12], it follows tht to ech regulr lnguge cn e uniquely ssocited miniml DFA generting it, nmely DFA with the fewest numer of sttes. Now, let L nd K e two regulr prefix-closed lnguges, where K represents the legl ehvior nd L represents the set of ll possile ehviors, including legl nd illegl ehvior. Let G L nd G K e the miniml DFA with generted lnguges L(G L ) nd L(G K ), respectively. Being such lnguges prefix-closed, mrked lnguges coincide with regulr lnguges. Strting from G L nd G K, we wnt to give procedure to construct unique DFA H where some sttes re good nd others re d. The strings terminting in good stte represent legl ehvior nd should elong to K. On the contrry, the strings terminting in d stte represent the foridden lnguge, i.e., should elong to L \ K. The min steps of the procedure to construct such DFA cn e summrized y Algorithm 1. The following property is stisfied y the DFA H uilt using the ove procedure. Proposition 3.9: Let H e the utomton uilt ccording to Algorithm 1, strting from two prefix-closed regulr lnguges K nd L. All strings tht finish in n unmrked stte re in K. All strings tht finish in mrked stte re in L \ K. Algorithm 1 Construction of the DFA H Let G K = (X,E,δ,x 0,X m ) e DFA where X, E, δ nd x 0 re the sme of G K nd X m =. Add new mrked stte to G K tht hs self-loop contining ll events in E. Add rcs leled E \ {e E δ(x,e)!} from ech stte x X to this new stte. Let H = G L G K e the utomton otined y the prllel composition of utomton G L nd utomton G K. Proof: Simply follows from the rules of construction of H using Algorithm 1. Note tht it cn never occur tht string finishes in n unmrked stte pssing through mrked stte. Indeed, y the rules of construction of H, if string reches mrked stte, ll events tht follow, never llow the stte to e chnged. Uniform q oservility cn e studied ccording to Algorithm 2. Algorithm 2 Uniform q-oservility Let X = {X \ X m } e the set of unmrked (good) sttes of H. while X = do Choose ritrrily one stte x X i 1. while i q do Compute the set of words of length i tht cn e generted y H strting from x. Prtition such words in equivlence clsses W j s. if some equivlence clss W : W K ut it is not W K then exit. {The lnguge K is not uniformly q oservle wrt L nd Σ i s}. else i = i + 1 end if end while X X \ {x} end while Exmple 3.10: Let L nd K e the two lnguges lredy considered in Exmple 3.5, nmely, L = ( + ) nd K = K 1 + K 2 where K 1 = ( ) nd K 2 = ( ). The DFA in Fig. 1 cn e otined pplying Algorithm 1 where G K is composed y x 0 nd x 1 while G L lso includes x 2. Therefore, ll strings strting from x 0 voiding x 2 elong to K. However, if string finishes in x 2 it elongs to L \K, i.e., it is d word. To study uniform 1-oservility we initilly ssume X = {x 0,x 1 }. Let us first focus on x 0. The set of words of unitry length strting from x 0 is {,}: nd oviously elong to different equivlence clsses, i.e., they re distinguishle, thus we continue the lgorithm. In prticulr, we repet the sme resoning for x 1 nd we conclude tht K is uniformly 381

1-oservle. Using similr rguments we conclude tht the lnguge K is not uniformly 2-oservle. x 1 x 2 x 3 c,, c IV. q DIAGNOSABILITY In this section we introduce new property, strictly relted to uniform q-oservility, tht we denote q-dignosility. Such property still concerns the possiility of estlishing if word given y the conctention of legl word w, nd word u on which we receive some informtion, is legl s well. The min difference of q-dignosility wrt q-oservility is on the informtion on u. We still ssume the presence of n oservers, ech one with its own lphet, nd coordintor. However, in the cse of q-dignosility oservtions re sent to the coordintion y single sites in the form of series of finite numer m of synchronized words, rther thn single word. Definition 4.1: Let Σ e finite lphet, nd Σ i Σ, with i = 1,...,n, e n su-lphets of Σ. Let L nd K e two prefix closed lnguges such tht K L Σ. The lnguge K is clled q dignosle wrt L nd Σ i, i = 1,...,n, if for ll m N nd sequence of m words (u 1,u 2,...,u m ) such tht u 1 u 2...u m L nd u i q, i = 1,...,m, it holds u 1 u 2...u m K f (P 1 (u 1 ),...,P n (u 1 ),...,...,P 1 (u m ),...,P n (u m )) = 1. (5) The notion of equivlence cn e esily extended to the cse of q dignosility. Definition 4.2: Let Σ e finite lphet, nd Σ i Σ, with i = 1,...,n, e n su-lphets of Σ. Let L nd K e two prefix closed lnguges such tht K L Σ. Consider two sequences of word (u 1,u 2,...,u m ) nd (v 1,v 2,...,v m ), where u 1 u 2 u m,v 1 v 2 v m L. The two sequences re dignosle equivlent, or simply equivlent, if P i (v j ) = P i (u j ) for ll i = 1,...,n nd ll j = 1,...,m. We denote this (u 1,u 2,...,u m ) (v 1,v 2,,v m ). Finlly, we sy tht two sequences tht re not equivlent re distinguishle. Oviously, if oth lnguges L nd K re regulr, y Algorithm 1 the nlysis of q dignosility cn e crried out using DFA where finl sttes correspond to d sttes, nd sequences tht terminte in them re not legl. Moreover, the following impliction holds. Proposition 4.3: If lnguge K is q dignosle wrt to lnguge L nd set of lphets Σ 1,...,Σ n, then it is lso q oservle wrt L nd Σ 1,...,Σ n. Proof: It is consequence of Definitions 3.2 nd 4.1. Indeed, consider ny word w K nd write it s w = u 1 u 2 u k where u i q for ll i. Then for ny word u w 1 L with u q we cn define function f w in Definition 3.2 in terms of function x 0 Fig. 2. c x 4 x 5 x 6 The DFA considered in Exmple 4.4 where q 7 is the d stte. f in Definition 4.1 s follows: f w (P 1 (u),...,p n (u)) = f (P 1 (u 1 ),...,P n (u 1 ),...,P 1 (u k ),...,P n (u k ), P 1 (u),...,p n (u)) showing tht K is uniformly q oservle wrt L nd Σ i s. On the contrry, q-oservility does not imply q- dignosility s shown y the following exmple. Although, the results presented ove hold for oth regulr nd non regulr lnguges, for the ske of simplicity the following exmple dels with regulr lnguges. Exmple 4.4: Let L e the lnguge generted y the DFA in Fig. 2 where x 0 is the initil stte, while K is the lnguge generted y the sme DFA with the sme initil stte, ut neglecting x 7, tht is the only d stte. Finlly, ssume three sites with lphets Σ 1 = {}, Σ 2 = {} nd Σ 3 = {c}, respectively. As shown in the following items, K is uniformly 3 oservle wrt L nd Σ i, i = 1,2,3. Let w = ε. All possile words u w 1 L with u = 3 finish in good sttes, without pssing through d stte. In prticulr, u 1 = termintes in x 3 nd u 2 = c in x 6. Therefore, it is f w (P 1 (u 1 ),P 2 (u 1 ),P 3 (u 1 )) = f w (,,ε) = 1 nd f w (P 1 (u 2 ),P 2 (u 2 ),P 3 (u 2 )) = f w (,,c) = 1. Let w =. Two possile sequence of length 3 my follow w, nmely u 3 = c w 1 K nd u 4 = w 1 K. However c, thus they cn e distinguished y the coordintor ssuming f w (P 1 (u 3 ),P 2 (u 3 ),P 3 (u 3 )) = f w (ε,,c) = 0 nd f w (P 1 (u 4 ),P 2 (u 4 ),P 3 (u 4 )) = f w (,,ε) = 1. Let w =. As in the ove item, there re two sequences of length 3 tht cn follow w, nmely u 5 = cc w 1 K nd u 6 = c w 1 K. However, these strings cn e distinguished eing cc c. Let w =. In this cse, there re 4 possile strings of length 3 tht my follow w, one in w 1 K, nmely u 7 =, the other three not in w 1 K, nmely, c, c nd cc. However, the word finishing in good stte cn e distinguished y ll words finishing in the d stte since it does not contin event c, while ll the others do. c x 7 382

Let w =. Also in this cse the good word (c) cn e distinguished y the d ones (cc,cc,ccc) since it only contins one c, while the others contin t lest two c. Let w =. Also in this cse the good word cn e distinguished y the d ones since it does not contin c, while ll the d do. Let w = c. The sme s in previous cse: if one c is oserved y Σ 3, the coordintor cn conclude tht the d stte x 7 is reched. Note tht no other words w need to e considered since the previous ones cover ll good sttes of the DFA. Moreover we do not need to consider words u of length smller thn 3 since y Proposition 3.6 uniform q oservility implies uniform (q 1)-oservility. Using similr rguments, we cn prove tht K is not 4 oservle. In prticulr, the two sequences c c my follow w = ε ut c w 1 K while c w 1 K. Thus no function f w my e defined to distinguish them. Finlly, let us prove tht even if K is 3 oservle, it is not 3 dignosle. Indeed, let us ssume u 1 =, u 2 = c, v 1 = nd v 2 = c. Since P 1 () = P 1 () =, P 2 () = P 2 () =, P 3 () = P 3 () = ε, P 1 (c) = P 1 (c) = ε, P 2 (c) = P 2 (c) = nd P 3 (c) = P 3 (c) = c then v 1 v 2 u 1 u 2. Being v 1 v 2 K nd u 1 u 2 K it will e impossile for the coordintor to distinguish mong them. The following result provides useful criterion to the nlysis of q-dignosility. Proposition 4.5: Let K e q oservle wrt given lnguge L nd set of lphets Σ i s. If fter ny ˆq q steps the stte is uniquely determined, q oservility = q dignosility. Proof: If fter ˆq q steps the stte is uniquely determined nd K is q oservle it is lwys possile to sy if the conctented word is in K. Using this rgument for finite numer of susequences, the sttement follows. Exmple 4.6: Let us consider gin the cse of Exmple 4.4 whose corresponding DFA H is tht reported in Fig. 2. As lredy proved K is not 3 dignosle even if it is 3 oservle. This result is consistent with Proposition 4.5. Indeed, if we consider u =, the first site oserves nd the second one oserves. Thus the current stte is not uniquely determined: oth x 2 nd x 5 re possile. We finlly present the following result. Proposition 4.7: Let us consider set of lphets Σ i, i = 1,...,n, such tht Σ 1... Σ n = Σ. Let K nd L e two prefix-closed lnguges such tht K L Σ. If K nd L re regulr lnguges, the q dignosility of K wrt L nd Σ i s is decidle for ny finite q N. Proof: We just give sketch of the proof. Since we re tking into ccount regulr lnguges we cn equivlently spek out DFA H constructed with Algorithm 1 with stte set X. To determine if the property holds for m = 1 we need to check ll words u 1 of length less thn or equl to q tht cn e generted y the DFA strting from the initil stte x 0. Consider the cse m = 2. After the first synchroniztion is performed, we do not know the current stte of the DFA ut we know it elongs to set X(u 1 ) = X(P 1 (u 1 ),...,P n (u 1 )) X nd the set Ξ 1 = {X(u 1 ) u 1 K, u 1 q} is finite. Now, for ll possile X 1 Ξ 1 we consider the the lnguge L(H X 1 ) = x X1 L(H x) where L(H x) denotes the lnguge generted y the utomton with initil stte x nd we need tho check ll words of length less thn or equl to q in this lnguge. As m is incresed one my hve lrger sets Ξ k to check ut eventully Ξ k = Ξ k+1 ecuse for ll k 1 it holds Ξ k 2 X. Hence there re t most 2 X lnguges L(H X k ) to consider nd the prolem is decidle. V. DYNAMIC OBSERVABILITY AND DIAGNOSABILITY In this section we focus on regulr prefix-closed lnguges nd consider prolem tht my occur in severl rel pplictions. We ssume tht the ctul stte of the system is known, nd we wnt to develop n lgorithm to determine the instnts t which it is necessry to synchronize the oservtions coming from the different sites, so tht the d stte is identified exctly s soon s it is reched. Oviously, the lst instnt t which synchroniztion occurs should e equl to the length of the shortest pth (denoted y k) from the ctul stte to d stte. Furthermore, ccording to Proposition 4.5, the stte in which the system is fter k steps should e uniquely determined such tht it is still possile to perform dignosis. The proposed lgorithm is lso sed on the following quite intuitive result. Proposition 5.1: Two consecutive synchroniztion performed fter q 1 nd q 2 steps, respectively, led to numer of consistent words/sttes smller thn unique synchroniztion fter q 1 + q 2 steps. Proof: Follows from the trivil considertion tht n intermedite dditionl synchroniztion cn only led to dditionl informtion, thus to reduced numer of consistent words/sttes. Let us now consider two regulr lnguges L nd K L generted y two DFA G L nd G K, respectively. Given n initil stte, Algorithm 3 computes the instnts t which it is necessry to synchronize to gurntee tht d stte is identified exctly in the instnt in which it is reched, nd in the cse tht no d stte is reched fter numer of steps equl to the length of the shortest pth from the current stte to d stte, the new stte is uniquely identified. Remrk 5.2: Algorithm 3 ensures tht the set of consistent sttes fter the lst synchroniztion t the k-th step is singleton, i.e., the ctul stte of the system is known fter the lst synchroniztion. This is trivil consequence of the fct, tht fter k steps ll equivlence clsses re singleton. On the contrry, the set of consistent sttes fter the intermedite synchroniztion is in generl not singleton. Exmple 5.3: Let us pply Algorithm 3 to the DFA in Fig. 3 ssuming Σ 1 = {}, Σ 2 = {} nd x 12 s the d stte. 383

Algorithm 3 Synchroniztion k length of the shortest pth from the ctul stte to d stte. Let I = {k} e the set of indices of steps t which we hve to synchronize. Compute ll words of length k nd split them in equivlence clsses W j with the sme projections on Σ i, i = 1,...,n. while W j 1 for ll j do Choose rndomly one word w W Compute n index p w such tht if new intermedite synchroniztion occurs fter p steps, w will not e equivlent to ny word in W \ { w}. I I {p}. Updte the set of equivlence clsses W j tking into ccount the new synchroniztion. end while x 11 x 10 Fig. 3. x 0 x 9 x 1 x 5 x 6 x 7 x 2 x 3 x 8 x 4 x 12, The DFA considered in Exmple 5.3 where x 12 is the d stte. The length of the shortest pth from x 0 to the d stte x 12 is k = 5. Hence, we intilly tke I = {5}. The set of strings of length k = 5 strting from x 0 is {,,,,}. Therefore, we cn define two equivlence clsses: W 1 = {,,} nd W 2 = {,}. In fct, P 1 () = P 1 () = P 1 () = nd P 2 () = P 2 () = P 2 () =. We rndomly choose n equivlence clss with crdinlity greter thn 1, e.g., W = W1. We rndomly choose w = nd consider p = 1. Thus w = ū v with ū = nd v =. Indeed, P 1 (ū) = while the projection of the first event of ll other sequences in W is equl to the empty string, thus the new synchroniztion mkes w not equivlent to ll the sequences in W \ { w}. Let I = {1,5}. The new equivlence clsses ssuming synchroniztion t steps 1 nd 5 re: W 1 = {}, W 1 = {,}, W 2 = {} nd W 2 = {}. We rndomly choose new equivlence clss of crdinlity greter thn 1, e.g., W = W 1. We rndomly choose w = nd consider p = 2. Let I = {1,2,5}. It is esy to verify tht the new equivlence clsses re singleton nd the lgorithm stops. Therefore, strting from x 0, in order to e le to uniquely identify the stte fter 5 steps (the length of the shortest pth to the d stte x 12 ) two dditionl synchroniztion should e performed. One fter one step, the second one fter one more step, nd the lst third one fter 3 further steps. Let us remrk tht this does not imply tht fter the two intermedite steps the stte is uniquely determined, while this is ensured fter the lst synchroniztion t step 5. At step 5, the lgorithm should e run gin considering s initil stte the new one tht hs een ctully reched fter the occurrence of 5 events. VI. CONCLUSIONS This pper dels with the prolem of estlishing if given ehvior is legl, sed on decentrlized oservtion performed y finite numer of sites, who re only le to oserve suset of the possile events. The sites trnsmit their oservtion to coordintor who tkes the decision concerning legcy of the occurred word. Two different properties hve een defined, nmely q oservility nd q dignosility, tht differ for the criterion used to synchronize the different sites. Finlly, n lgorithm to compute the instnts in which synchroniztion should occur, ssuming tht the initil stte is known, hs een given. It gurntees tht the occurrence of the n illegl word is detected s soon s it hs occurred. 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