Markig Estimatio of Petri Nets based o Partial Observatio Alessadro Giua ( ), Jorge Júlvez ( ) 1, Carla Seatzu ( ) ( ) Dip. di Igegeria Elettrica ed Elettroica, Uiversità di Cagliari, Italy {giua,seatzu}@diee.uica.it ( ) Dep. de If. e Ig. de Sistemas, Cetro Politécico Superior, Uiversidad de Zaragoza, Spai julvez@posta.uizar.es Abstract We preset a techique for estimatig the markig of a Petri et based o the observatio of trasitio labels. I particular, the mai cotributio of the paper cosists i derivig a methodology that ca hadle the case of odetermiistic trasitios, i.e., trasitios that share the same label. Uder some techical assumptios, the set of markigs cosistet with a observatio ca be represeted by a liear system with a fixed structure that does ot deped o the legth of the observed word. The validity of the proposed methodology is illustrated i detail through a umerical example. 1 Itroductio This paper deals with the problem of estimatig the markig of a Place/Trasitio (P/T) et based o the observatio of trasitio firigs. The problem of estimatig the state of a dyamic system is a fudametal issue i system theory ad the costructio of state observers for time-drive systems is treated i most liear systems textbooks. Although less popular i the case of discrete evet systems, the issue of state estimatio uder partial state observatio has bee discussed i the literature. For systems represeted as fiite automata, Ramadge [12] was the first to show how a observer could be desiged for a partially observed system. Caies et al. [2] showed how it is possible to use the iformatio cotaied i the past sequece of observatios (give as a sequece of observatio states ad cotrol iputs) to compute the set of cosistet states, while i [3] the observer output is used to steer the state of the plat to a desired termial state. A similar approach was also used by Kumar et al. [7] whe defiig observer based dyamic cotrollers i the framework of supervisory predicate cotrol problems. Özvere ad Willsky [10] proposed a approach for buildig observers that allows oe to recostruct the state of fiite automata after a word of bouded legth has bee observed, showig that a observer may have a expoetial umber of states. Let us defie the set of states cosistet with the observed behavior as the states i which the system may actually be give the observatio. There are two mai drawbacks i the above metioed automata based approaches to the desig of a discrete evet observer. Firstly, the set of cosistet states must explicitly be eumerated. Secodly, to compute the set of cosistet states at step k it is ot usually sufficiet to kow the ew observatio ad the set of cosistet states at step k 1, but it is ecessary to recompute this set as a fuctio of all previous observatios. Lookig for more efficiet approaches that do ot require the eumeratio of this set, we explored the possibility of usig Petri ets as discrete evet models [5, 6]. 1 Supported by a grat from D.G.A. REF B106/2001 0-7803-7896-2/03/$17.00 2003 IEEE 326 We showed that uder the followig three assumptios: (A1 ) the et structure is kow; (A2 ) the iitial markig is ot kow or is oly kow to belog to a iitial macromarkig, i.e., a give liear covex set; (A3 ) all trasitio firigs ca be observed; it is possible to represet the set of cosistet markigs (i.e., the states of the Petri et) as the solutio of a liear system that has a fixed structure which oly depeds o two parameters (the estimate ad the boud) that ca be recursively computed. Note that other authors [8] have also discussed the problem of estimatig the markig of a Petri et usig a mix of trasitio firig ad place observatios. I this paper, we further exted the approach of [5, 6] relaxig what we felt was its major limitatio, i.e., the assumptio (A3 ) that all trasitio firigs ca be observed. I fact, we assume that to each trasitio t is associated a label L(t) ad two or more trasitios may have the same label. Whe t fires, oly its label L(t) is observed ad this may itroduce odetermiism i the observer, i the sese that the observed word is ot sufficiet to recostruct the trasitio firig ad thus the actual markig. Note, however, that i this paper we restrict assumptio (A2 ) assumig that the iitial markig is perfectly kow. I effect, this may ot be strictly ecessary but we eed it i this paper to simplify the results we preset. I a first part of the paper, we show a rather simple result: usig the et state equatio it is possible to represet the set of cosistet markigs as the solutio of a liear system that ca be recursively computed, but whose structure, ufortuately, is ot fixed: it grows liearly with the legth of the observed word. A similar approach that uses a logical formalism rather tha liear programmig was also preseted by Beasser [1]. This author has studied the possibility of defiig the set of markigs reached firig a partially specified step of trasitios usig logical formulas, without havig to eumerate this set. I a secod part of the paper, we propose a differet approach that, uder some techical assumptios, allows us to characterize the set of cosistet markigs as the solutio of a differet liear system with a fixed structure that depeds o some parameters (the upper bouds u s) that ca be recursively computed. I particular, we make some restrictios o the structure of the labelig fuctio ad we assume that the same label caot be assiged to more tha two trasitios. Moreover, we assume thaodetermiistic trasitios (i.e., trasitios whose label is also associated to other trasitios) should also be cotact free, i.e., if t ad t are odetermiistic trasitios the set of iput ad output places of t caot itersect the set of iput ad output places of t. The validity of the proposed characterizatio has bee formally proved ad is illustrated i detail through a umerical example.
2 Backgroud o Petri ets I this sectio we recall the formalism used i the paper. For more details o Petri ets we address to [9]. A Place/Trasitio et (P/T et) is a structure N = (P, T, P re, P ost), where P is a set of m places; T is a set of trasitios; P re : P T N ad P ost : P T N are the pre ad post icidece fuctios that specify the arcs; C = P ost P re is the icidece matrix. The preset ad postset of a ode X P T are deoted X ad X while X = X X. A markig is a vector M : P N that assigs to each place of a P/T et a o egative iteger umber of tokes, represeted by black dots. We deote M(p) the markig of place p. A P/T system or et system N, M 0 is a et N with a iitial markig M 0. A trasitio t is eabled at M iff M P re(, t) ad may fire yieldig the markig M = M + C(, t). A labelig fuctio L : T E assigs to each trasitio t T a symbol from a give alphabet E. Note that the same label e E may be associated to more tha oe trasitio. Usig the otatio of [11] ad [4] we say that the labelig fuctio is λ-free. I the followig we say that a trasitio t is odetermiistic if its label is also associated to other trasitios, otherwise a trasitio t is said to be determiistic. We also deote T d the set of determiistic trasitios ad T the set of odetermiistic trasitios. Clearly, T = T d T. For simplicity of otatio, we assume that the trasitio eumeratio is such that T = {t j j = 1,, } ad T d = {t j j = + 1,, }, where = T. Aalogously, we say that a evet e is determiistic if there exists oly oe trasitio t such that L(t) = e, otherwise we say that the evet e is odetermiistic. Therefore, with o ambiguity o the otatio, we may write E = E d E. We deote as T e the set of trasitios labeled e, i.e, T e = {t T L(t) = e}. Moreover, we deote as s e {0, 1} the characteristic vector of T e, i.e., s e (i) = 1 if L(t i ) = e, ad s e (i) = 0 otherwise. We write M [σ M to deote that the eabled sequece of trasitios σ may fire at M yieldig M. We deote as w the word of evets associated to the sequece σ, i.e., w = L(σ). Moreover, we deote as σ 0 the sequece of ull legth ad w 0 the empty word. Fially, we use the otatio w i w to deote the geeric prefix of w of legth i k, where k is the legth of w. I particular, for i = 0, we have by defiitio the empty word, w 0 = ε. A markig M is reachable i N, M 0 iff there exists a firig sequece σ such that M 0 [σ M. The set of all markigs reachable from M 0 defies the reachability set of N, M 0 ad is deoted R(N, M 0 ). 3 Problem statemet I this paper we deal with the problem of estimatig the markig of a et system N, M 0 whose markig caot be directly observed. The followig properties of the system will be assumed. (A1) The structure of the et N is kow. (A2) The iitial markig M 0 is kow. (A3) Labels associated to trasitio firigs ca be observed. After the word w has bee observed, we defie the set C(w) of w-cosistet markigs as the set of all markigs i which the system may be give the observed behaviour. Defiitio 1. Give a observed word w, the set of w-cosistet markigs is C(w) = {M N m a sequece of trasitios σ : M 0 [σ M ad L(σ) = w}. t 5 c 2 p 1 t1 p 2 t 6 p 3 p 4 a d b t 3 p 5 t 2 p 6 t 7 p 7 t 4 p 8 a e b Figure 1: Petri et system that ca oly be partially observed Our goal is that of providig a systematic ad efficiet procedure to estimate the set of markigs that are cosistet with a observed word. Clearly, C(w 0 ) = M 0 ad C(w) is a sigleto if for all e i w, T e is a sigleto. O the cotrary, the degree of odetermiism may icrease as the cardiality of T e icreases. Fially, let us observe through a simple example, that the cardiality of the set of cosistet markigs may either icrease or decrease as the legth of the observed word icreases. Example 2. Let us cosider the Petri et system i figure 1 where T d = {t 5, t 6, t 7 } ad T = {t 1, t 2, t 3, t 4 }. More precisely, T a = {t 1, t 2 }, T b = {t 3, t 4 }, T c = {t 5 }, T d = {t 6 }, ad T e = {t 7 }. Clearly whe o evet has bee observed, C(ε) = {[1 0 1 0 1 0 1 1] T }. Let us first assume that the evet b is observed. Give the iitial markig M 0, either t 3 or t 4 may have bee fired, thus C(b) = {[1 0 0 1 1 0 1 1] T, [1 0 1 0 1 0 0 2] T }. Now, let a be the ext observed evet. Label a is associated to trasitios t 1 ad t 2 ad both trasitios are eabled at both markigs i C(b). Therefore, C(ba) = {[0 1 0 1 1 0 1 1] T, [1 0 0 1 0 1 1 1] T, [0 1 1 0 1 0 0 2] T, [1 0 1 0 0 1 0 2] T }. Now, if d is observed, we may be sure thaeither [1 0 0 1 0 1 1 1] T or [1 0 1 0 0 1 0 2] T i C(ba) may have bee reached because oe of these markigs eables t 6. Thus, C(bad) = {[0 0 1 1 1 0 1 1] T, [0 0 2 0 1 0 0 2] T }. If b is observed agai, both trasitios t 3 ad t 4 may have fired from the first markig i C(bad), while oly trasitio t 3 may have fired from the secod markig. Thus C(badb) = {[0 0 0 2 1 0 1 1] T, [0 0 1 1 1 0 0 2] T }. Fially, if we observe the determiistic evet c we ca coclude that oly the first markig i C(badb) is compatible with the last observatio, thus the actual markig of the et is completely recostructed ad C(badbc) = {[1 0 0 0 2 0 1 0] T }. 4 Computatio of the set of cosistet markigs We first preset a recursive algorithm strictly based o the defiitio of the set of cosistet markigs C(w), the we provide a algebraic characterizatio of C(w). Algorithm 3. 1. Let C(w 0 ) = M 0. 2. Let i = 0. 3. Wait util a ew evet e is observed. 4. Let i = i + 1. 5. Let w i = w i 1 e. 6. Let C(w i ) =. 7. For all M C(w i 1 ) do For all t such that M[t ad L(t) = e compute M = M + C(, t) ad let C(w i ) = C(w i ) M. 8. Goto 3. 327
Clearly, the mai disadvatage of the above iterative algorithm is that to compute the set of markigs that are cosistet with a observed word w of cardiality k, we prelimiary eed to compute the set of markigs that are cosistet with all prefixes w i w, i = 1,, k 1. A solutio to this problem cosists i usig a liear algebraic characterizatio of the set of cosistet markigs. Propositio 4. Let N, M 0 be a et system ad w = e 1,, e k be a observed word. The set of w-cosistet markigs is give by: C(w) = {M (k) N m 1 T σ (i) = 1 i = 1, k (a) s ei σ (i) = 1 i = 1, k (b) M (i 1) P re σ (i) i = 1, k (c) M (i) = M (i 1) + C σ (i) i = 1, k (d) σ (i) {0, 1} i = 1, k} (e) where M (0) = M 0 ad 1 is the -dimesioal colum vector of 1 s. Proof: It follows from the defiitio of the set of cosistet markigs. I fact, for ay observed evet e i, we itroduce a ukow vector σ (i) of zeros ad oes (costrait (e)) represetig the firig vector associated to the observed evet. The, the first costrait (a) imposes that whe the evet e i is observed, oly oe trasitio has fired ad the secod costrait (b) states that the label of that trasitio should be equal to the observed evet. Moreover, if a trasitio has fired, the it should be eabled by at least oe markig i the set C(w i 1 ) (iequality (c)) ad its firig brigs to a ew markig that is give by costrait (d). Example 5. Let us cosider agai the et system depicted i fig. 1. Let us assume that the observed evet is b. By virtue of Propositio 4 we may write: C(b) = {M (1) N 8 1 T σ (1) = 1, σ (1) 3 + σ (1) 4 = 1, M (0) P re σ (1), M (1) = M (0) + C σ (1), σ (1) {0, 1} 7 } where M (0) is the iitial markig. Now, let a be the ext observed evet. Usig Propositio 4 we may coclude that C(ba) = {M (2) N 8 1 T σ (1) = 1, σ (1) 3 + σ (1) 4 = 1, M (0) P re σ (1), M (1) = M (0) + C σ (1), σ (1) = {0, 1} 7, 1 T σ (1) = 1, σ (2) 1 + σ (2) 2 = 1, M (1) P re σ (2), M (2) = M (1) + C σ (2), σ (2) {0, 1} 7 } This example clearly shows that, eve if Propositio 4 eables us to directly describe the set of cosistet markigs without iteratig o the sets of markigs that are cosistet with the prefixes of the observed word, it still presets a sigificat drawback. I fact, both the umber of ukows ad the umber of costraits icrease as the legth of the observed word icreases. The mai goal of this paper is that of ivestigatig whether it is possible to defie the set of w-cosistet markigs usig a fixed (eve if large) umber of costraits. A geeral solutio to this problem has ot bee determied yet. But the wide variety of scearios we dealt with, eables us to coclude that this possibility is maily related to the degree of cotact of odermiistic trasitios ad to the umber of trasitios with the same label. Now, we derive some restrictive assumptios uder which it is possible to prove that the set of cosistet markigs may be expressed with a fixed umber of costraits. 5 The cotact-free case I this sectio we assume that the followig two coditios are verified: +1 +2 P r i t r e P r out +4 +5 +6 +3 P r+1 i t r+1 e P r+1 out Figure 2: The geeric couple of odetermiistic trasitios t r ad t r+1. (A4) for each label e E there are at most two trasitios such that L(t) = e, or equivaletly, T e 2; (A5) odetermiistic trasitios are cotact free, i.e., for ay two odetermiistic trasitios t i ad t j, it holds that t i t j =. Note that, give assumptio (A4), we always assume that the trasitio eumeratio is such that L(t r ) = L(t r+1 ) for r = 1, 3, 1. I the followig we formally prove that uder the above assumptios, a fixed umber of costraits, ot depedig o the legth of the observed word w, may be used to describe the set of w cosistet markigs. I particular, we formally prove that: C(w) = {M N m M = M 0 + C σ, σ r u r r = 1, 2,, (a) σ r + σ r+1 = r r = 1, 3,, 1 (b) (1) σ q = q q = + 1,, (c) σ N } (d) is the set of w cosistet markigs where the upper bouds u r s are appropriately computed ad r ( q ) deotes the umber of times a odetermiistic (determiistic) evet L(t r ) (L(t q )) has bee observed. Note that ay vector σ satisfyig costraits (a) to (d) of eq. (1) represets a admissible firig vector associated to a sequece of trasitios σ that may have fired ad whose labelig is equal to the observed word w, i.e., L(σ) = w. For ay couple of odetermiistic trasitios t r ad t r+1 we have 3 costraits: for each trasitio we eed a upper boud o the umber of times it may have fired, plus a additioal costrait keepig ito accout the total umber of times the correspodig odetermiistic evet L(t r ) = L(t r+1 ) has bee observed ( r ). O the cotrary, for each determiistic trasitio t q we oly eed oe costrait, because we exactly kow how may times it has fired. Lookig at hypothesis (A4) ad (A5) we may coclude that for each couple of odetermiistic trasitios, the ets we are dealig with cotai odetermiistic subets whose structure is like that oe show i fig. 2, where weights associated to arcs are ot required to be ordiary. I the followig page we have reported the algorithm that eables us to compute the upper bouds u r s used i eq. (1). The mai idea behid this algorithm is that of evaluatig the upper bouds u r s o the base of the kowledge of two parameters associated to odetermiistic trasitios. The first oe is zr i that represets the eablig degree of trasitio t r assumig that it has ever fired. This parameter is used to update the upper boud u r whe oe of the followig two cases occur. 328
Algorithm 6 (Upper bouds computatio). 1. Let u r = 0 for all r = 1,,. 2. Le q = 0 for all q = + 1,,. 3. Wait util a evet e is observed. 4. If e E d, the let t q be such that t q T d ad L(t q ) = e q = q + 1 if t q ( T ), the for every r {1,..., } such that t r ( t q ), do z i r = mi p t r { M0 (p) + t q p T d q P ost(p, t q ) t q p T } d q P re(p, t q ) u r = mi(u r, zr i ) edfor edif if t q (T ), the for every r {1,..., } such that t r ( t q ), do edfor edif 5. If e E the { zr out = max p t r u r = mi(u r, r z out P re(p, t r ) t p T d q P re(p, t q ) M 0 (p) t p T } d q P ost(p, t q ) P ost(p, t r ) r ) where r = r + 1 if r is odd, else r = r 1 for every r such that L(t r ) = e do { M0 (p) + zr i t = mi q p T d q P ost(p, t q ) t q p T } d q P re(p, t q ) p t r P re(p, t r ) u r = mi(u r + 1, z i r ) edfor edif 6. Goto 3. If a determiistic trasitio t q fires ad t q ( t r ) (see +1, +2 ad +3 i fig. 2), the value of zr i may decrease because we kow for sure that some toke(s) i Pr i were still available to eable t q. Thus, by defiitio of zr i, we may coclude that t r may have fired at most zr i times. A odetermiistic evet e is observed ad t r is a trasitio whose label is e. I such a case, the value of zr i keeps the same ad by defiitio of zr i we may coclude that t r may have fired at most zr i times. The secod parameter used to compute the upper bouds is zr out. It is a measure of the umber of tokes that have bee removed from the output places to t r by firig determiistic trasitios exitig Pr out (see +4, +5 ad +6 i fig. 2). I particular, the value of zr out is equal to the miimum umber of times trasitio t r has to be fired to fulfill the toke demads of the trasitios exitig Pr out. Cosequetly, it eables us to evaluate which is the maximum umber of times trasitio t r+1 may have fired, amely u r+1. Aalogously, the value of zr+1 out eables us to update the upper boud u r. Example 7. Let us cosider the ordiary Petri et system i figure 3. There are oly two odetermiistic trasitios whose label is a. The upper bouds u 1 ad u 2 may be updated as a cosequece of three differet types of observed evets. (1) If the first observed evet is a, the upper bouds should be both updated to u 1 = u 2 = 1 beig z i z i 1 = 2 = 2 ad the iitial bouds equal to zero. We are i the case of step 5 of Algorithm 6. (2) If a is observed agai, we are oce agai i the case of step 5 of Algorithm 6. The upper bouds are updated t 4 p 1 t 5 p 2 t 3 t 1 p 3 L(t 1 )=a p 4 t 6 t 7 t 8 p 5 t 2 L(t 2 )=a Figure 3: The Petri et system cosidered i example 7. to u 1 = u 2 = 2 beig z1 i = z2 i = 2 ad the previous bouds equal to oe. Now, let us assume that L(t 3 ) is observed, thus 3 = 1 ad z1 i = 1. This meas that for sure t 1 has fired at most oe time, otherwise t 3 would have ot bee eabled. Thus the upper boud of t 1 is updated to u 1 = 1. We are i the first if case of step 4 of Algorithm 6 beig t 3 a output trasitio to oe iput place of t 1. (3) Now, let us assume that L(t 8 ) is observed, thus w = aa L(t 3 ) L(t 8 ). This implies that t 1 should have fired at least oce, ad cosequetly t 2 should have fired at most oce. I fact, i such a case 8 = 1, z1 out = 1 ad cosequetly u 2 = 1. We are i the secod if case of step 4 of Algorithm 6. Lemma 8. Let us cosider a Petri et system N, M 0 ad let L : T E be its labelig fuctio. Assume that (A4) ad (A5) are satisfied. Let C(w) be defied as i equatio (1) where the upper bouds u r s are computed usig Algorithm 6. Assume that a label a is observed ad there is a trasitio t r labeled L(t r ) = a with boud u r such that it is disabled at ay markig i C(w). The the 329 p 6
ew boud u r computed by Algorithm 6 fulfills u r = u r. Proof: First, otice that if trasitio t r is disabled at ay markig i C(w) the all solutios of equatio (1) verify σ r = zr i where zr i is computed by Algorithm 6. I fact, σ r caot be greater tha zr i ad beig less would mea that there is a markig i C(w) i which t r is eabled. Furthermore σ r = u r sice if σ r < u r the there would exist aother solutio for equatio (1), let s say σ r, such that σ r > σ r, meaig that t r was eabled at the cosistet markig give by σ r. Therefore we have zr i = u r ad sice step 5 of Algorithm 6 computes u r as u r = mi(u r + 1, zr i ), we have u r = zr i = u r. Propositio 9. Let us cosider a Petri et system N, M 0 ad let L : T E be its labelig fuctio. Let us assume that assumptios (A4) ad (A5) are satisfied ad let w be a observed word of evets. The all markigs i the set C(w) defied as i equatio (1) are cosistet with the observed word w, whe the upper bouds u r s are computed usig Algorithm 6. Proof: We prove this by iductio o the legth of the observed word. Whe o evet is observed, i.e., whe w = w 0 is the empty word, usig equatio (1) we have that C(w 0 ) = {M 0 }, thus the statemet of the propositio holds. Moreover, whe a word w k 1 of legth k 1 is observed, we assume that all markigs i C(w k 1 ) are cosistet with w k 1, where C(w k 1 ) is defied as i equatio (1) ad the bouds are computed usig Algorithm 6. Now, let e be a ewly observed evet, ad let w = w k = w k 1 e. We have to prove that all markigs i C(w) are cosistet with the observed word w. For simplicity of presetatio i the followig we assume that there exists oly oe couple of odetermiistic trasitios, thus = 2 ad d = 2. We call a their label, i.e., L(t 1 ) = L(t 2 ) = a. Note that such a assumptio does ot affect the validity of the proof thaks to the cotact freeess hypothesis (A5). We partitio the set of trasitios as follows (see fig. 2): T = T T i T out T a (2) where T a = {t 1, t 2 }; P1 i (P1 out ) ad P2 i (P2 out ) are the set of iput (output) places to trasitios t 1 ad t 2 respectively. T i is the set of iput ad output trasitios to P1 i ad P2 i, apart from t 1 ad t 2 ; T out is the set of iput ad output trasitios to P1 out ad P2 out, apart from t 1 ad t 2 ; fially, T is the set of determiistic trasitios that are ot cotaied i the previous sets. Moreover, we defie the followig two sets 1 : σ 1 u 1 σ 1 u 1 σ S = 2 u 2 σ 1 + σ 2 = S σ = 2 u 2 a σ 1, σ 2 N σ 1 + σ 2 = (3) a σ 1, σ 2 N where S (S ) cosists of the subset of costraits of equatio (1) oly ivolvig the odetermiiistic trasitios t 1 ad t 2, whe the observed word is w k 1 (w). Clearly, these sets cotai the oly equatios that are related to the odetermiistic part of the et, thus oly a error o their defiitio may produce a error o the defiitio of the set of cosistet markigs. Therefore, the ext step of the iductio is proved if we demostrate that each solutio of S origiates from a solutio of S whe the bouds are updated usig Algorithm 6, i.e., 1 Slightly abusig the otatio, we deote with S ad S both the set of costraits give by (3) ad their respective solutios (σ 1, σ 2 ). if the observed evet is determiistic, i.e., e a, the S S; if the observed evet is odetermiistic, i.e., e = a, the give a solutio σ = (σ 1, σ 2 ) S, if t 1 (resp., t 2 ) is eabled from the markig correspodig to σ, the σ = (σ 1 + 1, σ 2 ) S (resp., (σ 1, σ 2 + 1) S ). Now, whe a evet e is observed, four differet cases may occur. (1) A trasitio t T has fired. I such a case S S ad the statemet of the propositio holds. (2) A trasitio t T i has fired. a. If t (P1 i ) (P2 i ), o boud is updated thus S S. b. If t (P1 i ) (P2 i ) the upper bouds may either stay the same or may be eve smaller thus S S. (3) A trasitio t T out has fired. a. If t (P1 out ) (P2 out ), o boud is updated thus S S. b. If t (P1 out ) (P2 out ) the upper bouds may either stay the same or may be eve smaller thus S S. (4) A trasitio t T a has fired. Let us deote Ta e the set of trasitios whose label is a ad that are eabled by at least oe markig i C(w k 1 ). Two differet cases may occur: (1) Ta e is a sigleto, i.e., either Ta e = {t 1 } or Ta e = {t 2 }. (2) Ta e = {t 1, t 2 }. 1. With o loss of geerality we may assume Ta e = {t 1 }. I such a case the geeric solutio (σ 1, σ 2) of S may always be writte as σ 1 = σ 1 + 1, σ 2 = σ 2. I fact, if this was ot possible, the σ 1 = 0 ad σ 2 = a = a + 1 > a u 2 = u 2, where the last equality follows from lemma 8. Therefore, we would obtai σ 2 > u 2, that leads to a cotradictio. Now, we wat to prove that ( σ 1, σ 2 ) is a solutio of S. By simply substitutig (σ 1, σ 2) i (3) where S is defied, ad takig ito accout tha a = a + 1, u 2 = u 2 ad u 1 = u 1 +1, we ca trivially verify that ( σ 1, σ 2 ) S. 2. Let us ow cosider the case i which Ta e = {t 1, t 2 }. We first observe that for at least oe trasitio t i Ta e, σ i > σmi i, where σi mi, i = 1, 2, is the miimum value of σ i for ay (σ 1, σ 2 ) S. I fact, if this was ot true, the for all solutios (σ 1, σ 2 ) S, ad (σ 1, σ 2) S it holds tha a = σ 1 +σ 2 = σ1 mi +σ2 mi σ 1 +σ 2 = a cotradictig a = a + 1 > a. Now, with o loss of geerality we assume that σ 1 > σ mi 1 0. The, we may write σ 1 = σ 1 1 ad σ 2 = σ 2. We show that ( σ 1, σ 2 ) S. The oly costrait that is ot trivially verified is σ 2 u 2. I fact, σ 2 u 2 σ 2 u 2. However, we show that if σ 2 = u 2 = u 2 +1 the σ 1 = a u 2 = a +1 u 2 1 = a u 2. By assumptio σ 1 > σ mi 1, thus σ 1 > a u 2 that leads to a cotradictio. Propositio 10. Let us cosider a et system N, M 0 ad let L : T E be its labelig fuctio. Let us assume that assumptios (A4) ad (A5) are satisfied ad let w be a observed word of evets. The all markigs that are cosistet with the observed word w are cotaied i C(w), whe C(w) is defied as i equatio (1) ad the upper bouds u r s are computed usig Algorithm 6. Proof: We prove this by iductio o the legth of the observed word. Clearly, whe o evet is observed the oly cosistet markig is the iitial oe, thus the statemet of the propositio holds. Moreover, we assume that it also holds whe a word w k 1 is observed, i.e., we assume that there exists o markig that is cosistet with w k 1 ad that is ot cotaied i C(w k 1 ). 330
To complete the prove, we must demostrate that whe a ew evet e is observed, i.e., whe the curret word is w = w k = w k 1 e, all markigs that are cosistet with w are cotaied i C(w). As i the case of the previous propositio, thaks to the cotact freeess assumptio (A5), we may assume that there exists oly oe couple of odetermiistic trasitios, amely t 1 ad t 2. Therefore, we may restrict our attetio to the sets S ad S defied i equatio (3). Now, the ext step of the iductio is proved if we demostrate that, from each solutio (σ 1, σ 2 ) S correspodig to a markig i C(w k 1 ) eablig a trasitio labeled e, we get a solutio (σ 1, σ 2) S that is a cosistet markig associated to the observatio of e. We refer agai to the partitio of T itroduced via equatio (2) ad we cosider four differet cases. (1) A trasitio t T fires. Beig S S, the statemet of the propositio is trivially verified. (2) A trasitio t T i fires. I such a case, S S ad we must prove that whe updatig the bouds we are oeglectig markigs that are cosistet with w. However, by lookig at Algorithm 6 we may observe that S S if ad oly if r {1, 2} such that t ( t r ) ad < u r (first if case of step 4 of Algorithm 6). But this is correct because if we allow u r to be greater tha zr i, the o egativity costraits would be violated. (3) A trasitio t T out fires. This case is similar to the previous oe. I fact, S S. I particular, S S if z i r ad oly if r {1, 2} such that t (t r) ad r zr out < u r, where r is defied as i step 4 of Algorithm 6. But this is correct, because zr out deotes by defiitio the umber of times trasitio t r has fired for sure. If we allow u r to be greater tha r zr out (or equivaletly u r to be smaller tha zr out ), the o egativity costraits are violated. (4) A trasitio t T a fires. We must prove that, give a solutio σ = (σ 1, σ 2 ) S, if t 1 (resp., t 2 ) is eabled from the markig correspodig to σ, the σ = (σ 1 +1, σ 2 ) S (resp., (σ 1, σ 2 + 1) S ). With o loss of geerality we may assume that t 1 is eabled from the markig correspodig to σ. This implies that for that σ it holds that σ 1 < zr i beig by defiitio zr i the eablig degree of trasitio t r assumig that t r has ever fired. Thus, σ 1 < zr i, σ 1 u r = σ 1 = σ 1 + 1 mi(u r + 1, zr i ) = u 1. Moreover, σ 1 1 + σ 2 = a σ 1 + σ 2 = a. Therefore, we may coclude that (σ 1, σ 2) S. Theorem 11. Let us cosider a et system N, M 0 ad let L : T E be its labelig fuctio. Let us assume that assumptios (A4) ad (A5) are satisfied ad let w be a observed word of evets. The the set C(w) defied by equatio (1) cotais all ad oly those markigs that are cosistet with the observed word w, whe the upper bouds u r s are computed usig Algorithm 6. Proof: It follows from propositios 9 ad 10. Example 12. Let us cosider agai the Petri et system i fig. 1. Assumptios (A4) ad (A5) are verified. Thus, by virtue of Theorem 11, the set of cosistet markigs ca be described i terms of equatio (1) where the upper bouds are computed usig Algorithm 6. All bouds are iitially set to zero, thus C(ε) = {M N 8 M = M 0 + C σ, σ 1, σ 2, σ 3, σ 4 0, σ 1 + σ 2 = 0, σ 3 + σ 4 = 1, σ 5 = σ 6 = σ 7 = 0, σ N 7 } ad the oly admissible firig vector is σ = 0. Assume that b is observed. Both u 3 ad u 4 are updated to oe, while the other bouds keeps equal to zero. Thus, C(b) = {M N 8 M = M 0 + C σ, σ 1 0, σ 2 0, σ 3 1, σ 4 1, σ 1 + σ 2 = 0, σ 3 + σ 4 = 1, σ 5 = σ 6 = σ 7 = 0, σ N 7 }. It is easy to verify that i this case there are two admissible firig vectors ad C(b) = {[1 0 0 1 1 0 1 1] T, [1 0 1 0 1 0 0 2] T }. Similarly, if a is observed, we get u 1 = u 2 = 1 ad C(ba) = {M N 8 M = M 0 +C σ, σ 1 1, σ 2 1, σ 3 1, σ 4 1, σ 1 + σ 2 = 1, σ 3 + σ 4 = 1, σ 5 = σ 6 = σ 7 = 0, σ N 7 }. This implies that there are four admissible firig vectors ad C(ba) = {[0 1 0 1 1 0 1 1] T, [1 0 0 1 0 1 1 1] T, [0 1 1 0 1 0 0 2] T, [1 0 1 0 0 1 0 2] T }. Now, if d is observed, we have that z1 out = 1. Cosequetly u 2 = 0 ad C(bad) = {M N 8 M = M 0 + C σ, σ 1 1, σ 2 0, σ 3 1, σ 4 1, σ 1 + σ 2 = 1, σ 3 + σ 4 = 1, σ 6 = 1, σ 5 = σ 7 = 0, σ N 7 }. Fially, if the whole observed word is w = badbc, the the markig is perfectly kow beig σ = [1 0 2 0 1 1 0] T the oly admissible firig vector. 6 Coclusios We have preseted a markig estimatio procedure that ca be applied to labeled Petri ets. Uder some assumptios, we proved that the markigs cosistet with a observed sequece ca be described by a costrait set of liear iequalites: this set has a fixed structure that does ot chage as the legth of the observed sequece icreases. Refereces [1] A. Beasser, Reachability i Petri ets: a approach based o costrait programmig (i Frech), Ph.D. Thesis, Uiversité de Lille 1, 2000. [2] P.E. Caies, R. Greier, S. 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