Verification of Detectability Using Petri Nets and Detector
11 Verification of Detectability Using Petri Nets andDetector
Hao Lan, Yin Tong,
Member, IEEE
Jin Guo and Carla Seatzu,
Senior Member, IEEE
Abstract —Detectability describes the property of a systemto uniquely determine, after a finite number of observations,the current and subsequent states. In this paper, to reducethe complexity of checking the detectability properties in theframework of bounded labeled Petri nets, we use a new tool,which is called detector, to verifying the strong detectability andperiodically strong detectability. First, an approach, which isbased on the reachable graph and its detector, is proposed. Then,we develop a novel approach which is based on the analysis ofthe detector of the basis reachability graph. Without computingthe whole reachability space, and without building the observer,the proposed approaches are more efficient.
Index Terms —Detectability, Petri nets, Detector, Discrete eventsystems.
I. I
NTRODUCTION
In recent years, detectability has drawn a lot of attentionfrom researchers in the discrete event system (DES) com-munity [1], [2], [3], [4], [5]. This property characterizes theability of a system to determine the current and the subsequentstates of the system after the observation of a finite numberof events. Detectability has been studied earlier in DES withanother terminology, which is called “observability” [6], [7],[8]. The observability of the current state and initial stateare discussed in [6], and whether the current state can bedetermined periodically is investigated in [7]. The property ofdetectability in DESs has been studied systematically in theliterature [1], [3], [9], [10], [11]. The notion of detectabilitywas first proposed and studied in [9] in the deterministicfinite automaton framework based on the assumption that thestates and the events are partially observable. Shu et al. [9]defined four types of detectability: strong detectability, weakdetectability, strong periodic detectability, and weak periodicdetectability. And the four types of detectability are verifiedby an approach whose complexity is exponential with respectto the number of states of the system. Polynomial algorithmsto check strong detectability and strong periodic detectabilityof an automaton have been proposed in [10]. While checkingweak detectability and weak periodic detectability is provedto be PSPACE-complete and that PSPACE-hardness [1], evenfor a very restricted type of automata [3]. The notation ofdetectability is also extended to delayed DESs [12], modularDESs [13] and stochastic DESs [2], [11], and the enforcementof the detectability is proposed in [14], [15].
H. Lan, Y. Tong (Corresponding Author), and Jin Guo are withthe School of Information Science and Technology, Southwest JiaotongUniversity, Chengdu 611756, China [email protected];[email protected]; [email protected]
C. Seaztu is with the Department of Electrical and Electronic Engineering,University of Cagliari, 09123 Cagliari, Italy [email protected]
Petri nets are widely used to model many classes of con-current systems, some problems such as supervisory control[16], fault diagnosis [17], opacity [18], etc. can be solved moreefficiently in Petri nets. The detectability of unlabeled Petrinets was proposed by Giua and Seatzu [8], including markingobservability and strong marking observability. In [19], theauthors extend strong detectability and weak detectability inDESs to labeled Petri nets. Strong detectability is proved tobe decidable and checking the property is EXPSPACE-hard,while weak detectability is proved to be undecidable. In ourprevious work, we first extend the four detectability propertiesto labeled Petri nets in [4], then we relax detectability to C-detectability that only requires that a given set of crucial statescan be distinguished from other states [5]. In [4], it is shownthat detectability can be efficiently verified by using Petri nets.However, this method requires the construction of an observerof the basis reachability graph (BRG) of the LPN system.Since in the worst case, the complexity of constructing theobserver is exponential to the number of states of the BRG.Thus, it is important to search for more efficient algorithmsfor checking detectability in labeled Petri nets.In this paper, we develop a method to check strong de-tectability and periodically strong detectability with lowercomplexity, compared with [4]. By assumption that the initialstate of the observed behavior is not known and the systemsevolution is only partially observed, the method is based onthe construction of a new tool, called “detector”, which wasfirst proposed in [10] for verification of detectability in theframework of automation. We present necessary and sufficientconditions for the strong detectability and periodically strongdetectability, by analyzing the detector of the BRG of theoriginal LPN system. Thanks to basis markings and detector,there is no need to enumerate all the markings and no needto build the observer. This leads to a relevant advantage interms of computational complexity since the basis reachabilitygraph (BRG) is usually much smaller than the RG and thecomplexity of the detector is polynomial time. Further more,rather than computing all cycles in the detector [10], [12],[20], which is NP-complete, we show that detectability can beverified with polynomial complexity.The rest of the paper is organized as follows. In Section II,background on labeled Petri nets, basis markings and thedefinition of four detectability properties is provided. Basedon the RG and its detector, we propose an approach to verifythe strong detectability, periodically strong detectability inSection III. In Section IV, the efficient approaches to verifythe strong detectability, periodically strong detectability arepresented. Conclusions are finally drawn in Section V where a r X i v : . [ ee ss . S Y ] A ug our future lines of research in this framework are illustrated.II. P RELIMINARIES AND B ACKGROUND
In this section we recall the formalisms used in the paperand some results on state estimation in Petri nets. For moredetails, we refer to [17], [21], [22].
A. Petri Nets A Petri net is a structure N = ( P, T, P re, P ost ) , where P is a set of m places , graphically represented by circles; T is a set of n transitions , graphically represented by bars; P re : P × T → N and P ost : P × T → N are the pre- and post-incidence functions that specify the arcs directed fromplaces to transitions, and vice versa. The incidence matrix ofa net is denoted by C = P ost − P re . A Petri net is acyclic if there are no oriented cycles.A marking is a vector M : P → N that assigns to eachplace a non-negative integer number of tokens, graphicallyrepresented by black dots. The marking of place p is denotedby M ( p ) . A marking is also denoted by M = (cid:80) p ∈ P M ( p ) · p .A Petri net system (cid:104)
N, M (cid:105) is a net N with initial marking M .A transition t is enabled at marking M if M ≥ P re ( · , t ) and may fire yielding a new marking M (cid:48) = M + C ( · , t ) .We write M [ σ (cid:105) to denote that the sequence of transitions σ = t j · · · t jk is enabled at M , and M [ σ (cid:105) M (cid:48) to denote that the fir-ing of σ yields M (cid:48) . The set of all enabled transition sequencesin N from marking M is L ( N, M ) = { σ ∈ T ∗ | M [ σ (cid:105)} . Givena transition sequence σ ∈ T ∗ , the function π : T ∗ → N n associates with σ the Parikh vector y = π ( σ ) ∈ N n , i.e., y ( t ) = k if transition t appears k times in σ . Given a sequenceof transitions σ ∈ T ∗ , its prefix , denoted by σ (cid:48) (cid:22) σ , is a stringsuch that ∃ σ (cid:48)(cid:48) ∈ T ∗ : σ (cid:48) σ (cid:48)(cid:48) = σ . The length of σ is denotedby | σ | .A marking M is reachable in (cid:104) N, M (cid:105) if there exists a tran-sition sequence σ such that M [ σ (cid:105) M . The set of all markingsreachable from M defines the reachability set of (cid:104) N, M (cid:105) ,denoted by R ( N, M ) . A Petri net system is bounded if thereexists a non-negative integer k ∈ N such that for any place p ∈ P and any reachable marking M ∈ R ( N, M ) , M ( p ) ≤ k holds.A labeled Petri net (LPN) system is a 4-tuple G = ( N, M ,E, (cid:96) ) , where (cid:104) N, M (cid:105) is a Petri net system, E is the alphabet (a set of labels) and (cid:96) : T → E ∪ { ε } is the labeling function that assigns to each transition t ∈ T either a symbol from E or the empty word ε . Therefore, the set of transitionscan be partitioned into two disjoint sets T = T o ˙ ∪ T u , where T o = { t ∈ T | (cid:96) ( t ) ∈ E } is the set of | T o | = n o observabletransitions and T u = T \ T o = { t ∈ T | (cid:96) ( t ) = ε } is the setof | T u | = n u unobservable transitions. The labeling functioncan be extended to transition sequences (cid:96) : T ∗ → E ∗ as (cid:96) ( σt ) = (cid:96) ( σ ) (cid:96) ( t ) with σ ∈ T ∗ and t ∈ T .Given a set of markings Y ⊆ R ( N, M ) , the languagegenerated by G from Y is L ( G, Y ) = (cid:83) M ∈ Y { w ∈ E ∗ |∃ σ ∈ L ( N, M ) : w = (cid:96) ( σ ) } . In particular, the lan-guage generated by G from R ( N, M ) is L ( G, R ( N, M )) = (cid:83) M ∈ R ( N,M ) { w ∈ E ∗ |∃ σ ∈ L ( N, M ) : w = (cid:96) ( σ ) } that is simply denoted by L ( G ) . Let w ∈ L ( G ) be an observed word.We denote as C ( w ) = { M ∈ N m |∃ M (cid:48) ∈ R ( N, M ) , σ ∈ L ( N, M (cid:48) ) : M (cid:48) [ σ (cid:105) M, (cid:96) ( σ ) = w } . (1)the set of markings consistent with w . When |C ( w ) | (cid:54) = 1 ,markings in C ( w ) are confusable since any of them couldbe the current marking of the system. Correspondingly, wedenote as L ( G ) = { σ ∈ T ∗ |∃ M ∈ R ( N, M ) : M [ σ (cid:105)} the setof transition sequences enabled at a marking in R ( N, M ) .Finally we denote as L ω ( G ) = { σ ∈ T ∗ | σ ∈ L ( G ) ∧| σ | is infinite } the set of transition sequences of infinite lengththat are enabled at some markings in R ( N, M ) .Given an LPN system G = ( N, M , E, (cid:96) ) and the set ofunobservable transitions T u , the T u -induced subnet N (cid:48) =( P, T (cid:48) , P re (cid:48) , P ost (cid:48) ) of N , is the net resulting by removingall transitions in T \ T u from N , where P re (cid:48) and
P ost (cid:48) are the restriction of
P re , P ost to T u , respectively. Theincidence matrix of the T u -induced subnet is denoted by C u = P ost (cid:48) − P re (cid:48) . B. Basis Markings
In this subsection we review the notion and some results ofbasis markings, which is proposed in [17], [18], [23].
Definition 2.1:
Given a marking M and an observabletransition t ∈ T o , we denote as Σ( M, t ) = { σ ∈ T ∗ u | M [ σ (cid:105) M (cid:48) , M (cid:48) ≥ P re ( · , t ) } the set of explanations of t at M and Y ( M, t ) = { y u ∈ N n u |∃ σ ∈ Σ( M, t ) : y u = π ( σ ) } the set of e -vectors . (cid:5) After firing any unobservable transition sequence in Σ( M, t ) at M , the transition t is enabled. To provide a compactrepresentation of the reachability set, we are interested infinding the explanations whose firing vector is minimal. Definition 2.2:
Given a marking M and an observabletransition t ∈ T o , we denote as Σ min ( M, t ) = { σ ∈ Σ( M, t ) | (cid:64) σ (cid:48) ∈ Σ( M, t ) : π ( σ (cid:48) ) (cid:8) π ( σ ) } the set of minimal explanations of t at M and Y min ( M, t ) = { y u ∈ N n u |∃ σ ∈ Σ min ( M, t ) : y u = π ( σ ) } as thecorresponding set of minimal e -vectors . (cid:5) There are many approaches to calculate Y min ( M, t ) . In par-ticular, Cabasino et al present an approach that only requiresalgebraic manipulations when the T u -induced subnet is acyclic[17]. Definition 2.3:
Given an LPN system G = ( N, M , E, (cid:96) ) whose T u -induced subnet is acyclic, its basis marking set M b is defined as follows: • M ∈ M b ; • If M ∈ M b , then ∀ t ∈ T o , y u ∈ Y min ( M, t ) , M (cid:48) = M + C ( · , t ) + C u · y u ⇒ M (cid:48) ∈ M b . A marking M b ∈ M b is called a basis marking of G . (cid:5) The set of basis markings contains the initial marking andall other markings that are reachable from a basis marking byfiring a transition sequence σ u t , where t ∈ T o is an observable t (b) p t ( c ) t ( b )p t ( c ) t (a)p p p p t ( a ) t ( a ) t ( b ) p p a p p cp p ba a bb c p p ,p a p ,p p ,p cp p ,p b a a b b cp t (b)p p t ( a ) t ( a )t (a) p p bp a a a X p ,p a p ,p p ,p a a p p b b ba a a at (a) p p t ( ε )t ( a ) p p t ( b )t ( ε ) t ( c )p p M =p +p M =p M =p +p M =p M =p +p t ( ε ) t ( a ) t (a) t ( b )t ( ε ) t ( c ) (M ,M ) (M ,M ,M ) (M )(M ) a ac b b all {M }b {M ,M ,M } {M }ac bc a b all {M } b {M ,M } {M }a c b c ab{M ,M } {M ,M }a a bc a aall {(M ,0)}b {(M ,0),(M ,1)}{(M ,0)} a c bc a b c t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p t ( a ) t ( a )t (a) t ( b )t ( ε )t ( c ) (M ) (M ,M ,M ) (M ) (M ) a ac b(M ) aall {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)}a c bcb {(M ,0),(M ,1)} c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0)} {(M ,0)}aa bc b c a bbcall {M } b {M ,M ,M ,M }{M }a c bc b {M ,M ,M }c a ba all {M ,M } {M ,M } {M ,M }{M ,M }{M ,M }{M ,M } {M }{M } a a a aa a aa aa aa bbb b ccc c b b{(M ,0)} {(M ,0)}{(M ,1)}{(M ,0)} t ( a ) t ( a ) t (a) t ( b ) t ( c ) t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p p t ( ε ) M =p t (a) t ( ε )t ( a ) t ( b ) t ( a ) t ( c ) t ( ε )all {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)} a c bcb {(M ,0),(M ,1)}c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)}{(M ,0),(M ,1)} {(M ,0)}{(M ,0)}aa bc b c a b bc all {M }b {M ,M ,M ,M }{M } a c bc b {M ,M ,M }c a ba all {M ,M }{M ,M }{M ,M }{M ,M }{M ,M } {M ,M } {M } {M } a a aaaa a a aa a a bbb b c c c c b (M ,0) (M ,0) (M ,1) (M ,1) t ( a ) t ( a )t (a) t ( b ) t ( c ) Fig. 1. The LPN system in Example 2.4. transition and π ( σ u ) = y u is a minimal explanation of t at M . Clearly, M b ⊆ R ( N, M ) , and in practical cases thenumber of basis markings is much smaller than the numberof reachable markings [17], [23]. And the number of basismarkings is finite if the corresponding LPN system is bound.We denote as C b ( w ) = M b ∩ C ( w ) the set of basis markingscorresponding to a given observation w ∈ L ( G ) . Example 2.4:
Let us consider the LPN system in Fig. 1,where T o = { t , t , t , t , t } , T u = { t , t } . Transitions t , t and t are labeled by a , transition t is labeled by b , and transition t is labeled by c . At the initial marking M = [1 0 0 0 0 0 0] T , the set of minimal explanations of t is Σ min ( M , t ) = { t } , and thus Y min ( M , t ) = { [1 0] T } .The corresponding basis marking is M + C ( · , t ) + C u · [1 0] T = M = [0 0 1 1 0 0 0] T . At M , the set ofminimal explanations of t is Σ min ( M , t ) = { ε } , andthus Y min ( M , t ) = { (cid:126) } . The corresponding basis markingobtained is M + C ( · , t ) + C u · (cid:126) M = [0 0 0 0 0 0 1] T . (cid:5) Proposition 2.5: [5], [17] Let G = ( N, M , E, (cid:96) ) be anLPN whose T u -induced subnet is acyclic, M b ∈ M b a basismarking of G , and w ∈ L ( G ) an observation generated by G .We have1) A marking M is reachable from M b if and only if M = M b + C u · y u (2)has a nonnegative solution y u ∈ N n u .2) C ( w ) = (cid:91) M b ∈C b ( w ) { M ∈ N m | M = M b + C u · y u : y u ∈ N n u } Statement 1) of Proposition 2.5 implies that any solution y u ∈ N n u of Eq. (2) corresponds to the firing vector of afirable transition sequence σ from M b , i.e., M b [ σ (cid:105) and π ( σ ) = y u . According to Statement 2), the set of markings consistentwith an observation can be characterized using linear algebrawithout an exhaustive marking enumeration. C. Detectability
In this subsection we recall the definitions of the detectabil-ity problems of the LPN system. We assume that the initialmarking M of the LPN system is given, but the observationcould be generated from any marking in R ( N, M ) . As in [4],we make the following two assumptions: 1) the LPN system G is deadlock free. 2) the T u -induced subnet of G is acyclic.For more details, we refer to [4]. t (b) p t ( c ) t ( b )p t ( c ) t (a)p p p p t ( a ) t ( a ) t ( b ) p p a p p cp p ba a bb c p p ,p a p ,p p ,p cp p ,p b a a b b cp t (b)p p t ( a ) t ( a )t (a) p p bp a a a X p ,p a p ,p p ,p a a p p b b ba a a at (a) p p t ( ε )t ( a ) p p t ( b )t ( ε ) t ( c )p p M =p +p M =p M =p +p M =p M =p +p t ( ε ) t ( a ) t (a) t ( b )t ( ε ) t ( c ) (M ,M ) (M ,M ,M ) (M )(M ) a ac b b all {M }b {M ,M ,M } {M }ac bc a b all {M } b {M ,M } {M }a c b c ab{M ,M } {M ,M }a a bc a aall {(M ,0)}b {(M ,0),(M ,1)}{(M ,0)} a c bc a b c t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p t ( a ) t ( a )t (a) t ( b )t ( ε )t ( c ) (M ) (M ,M ,M ) (M ) (M ) a ac b(M ) aall {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)}a c bcb {(M ,0),(M ,1)} c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0)} {(M ,0)}aa bc b c a bbcall {M } b {M ,M ,M ,M }{M }a c bc b {M ,M ,M }c a ba all {M ,M } {M ,M } {M ,M }{M ,M }{M ,M }{M ,M } {M }{M } a a a aa a aa aa aa bbb b ccc c b b{(M ,0)} {(M ,0)}{(M ,1)}{(M ,0)} t ( a ) t ( a ) t (a) t ( b ) t ( c ) t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p p t ( ε ) M =p t (a) t ( ε )t ( a ) t ( b ) t ( a ) t ( c ) t ( ε )all {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)} a c bcb {(M ,0),(M ,1)}c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)}{(M ,0),(M ,1)} {(M ,0)}{(M ,0)}aa bc b c a b bc all {M }b {M ,M ,M ,M }{M } a c bc b {M ,M ,M }c a ba all {M ,M }{M ,M }{M ,M }{M ,M }{M ,M } {M ,M } {M } {M } a a aaaa a a aa a a bbb b c c c c b (M ,0) (M ,0) (M ,1) (M ,1) t ( a ) t ( a )t (a) t ( b ) t ( c ) (a) t (b) p t ( c ) t ( b )p t ( c ) t (a)p p p p t ( a ) t ( a ) t ( b ) p p a p p cp p ba a bb c p p ,p a p ,p p ,p cp p ,p b a a b b cp t (b)p p t ( a ) t ( a )t (a) p p bp a a a X p ,p a p ,p p ,p a a p p b b ba a a at (a) p p t ( ε )t ( a ) p p t ( b )t ( ε ) t ( c )p p M =p +p M =p M =p +p M =p M =p +p t ( ε ) t ( a ) t (a) t ( b )t ( ε ) t ( c ) (M ,M ) (M ,M ,M ) (M )(M ) a ac b b all {M }b {M ,M ,M } {M }ac bc a b all {M } b {M ,M } {M }a c b c ab{M ,M } {M ,M }a a bc a aall {(M ,0)}b {(M ,0),(M ,1)}{(M ,0)} a c bc a b c t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p t ( a ) t ( a )t (a) t ( b )t ( ε )t ( c ) (M ) (M ,M ,M ) (M ) (M ) a ac b(M ) aall {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)}a c bcb {(M ,0),(M ,1)} c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0)} {(M ,0)}aa bc b c a bbcall {M } b {M ,M ,M ,M }{M }a c bc b {M ,M ,M }c a ba all {M ,M } {M ,M } {M ,M }{M ,M }{M ,M }{M ,M } {M }{M } a a a aa a aa aa aa bbb b ccc c b b{(M ,0)} {(M ,0)}{(M ,1)}{(M ,0)} t ( a ) t ( a ) t (a) t ( b ) t ( c ) t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p p t ( ε ) M =p t (a) t ( ε )t ( a ) t ( b ) t ( a ) t ( c ) t ( ε )all {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)} a c bcb {(M ,0),(M ,1)}c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)}{(M ,0),(M ,1)} {(M ,0)}{(M ,0)}aa bc b c a b bc all {M }b {M ,M ,M ,M }{M } a c bc b {M ,M ,M }c a ba all {M ,M }{M ,M }{M ,M }{M ,M }{M ,M } {M ,M } {M } {M } a a aaaa a a aa a a bbb b c c c c b (M ,0) (M ,0) (M ,1) (M ,1) t ( a ) t ( a )t (a) t ( b ) t ( c ) (b)Fig. 2. The RG of the LPN system in Fig. 1 (a), the observer of the RG (b). Definition 2.6: [ Strong detectability ] An LPN system G =( N, M , E, (cid:96) ) is strongly detectable if ∃ K ∈ N , ∀ σ ∈ L ω ( G ) , ∀ σ (cid:48) (cid:22) σ, | w | ≥ K ⇒ |C ( w ) | = 1 , where w = (cid:96) ( σ (cid:48) ) . (cid:5) In words, an LPN system is strongly detectable if the currentand the subsequent states of the system can be determined aftera finite number of events observed for all trajectories of thesystem.
Definition 2.7: [ Weak detectability ] An LPN system G =( N, M , E, (cid:96) ) is weakly detectable if ∃ K ∈ N , ∃ σ ∈ L ω ( G ) , ∀ σ (cid:48) (cid:22) σ, | w | ≥ K ⇒ |C ( w ) | = 1 , where w = (cid:96) ( σ (cid:48) ) . (cid:5) In simple words, an LPN system is weakly detectableif we can determine, after a finite number of observations,the current and subsequent states of the system for sometrajectories of the system.
Definition 2.8: [ Periodically strong detectability ] An LPNsystem G = ( N, M , E, (cid:96) ) is periodically strongly detectable if ∃ K ∈ N , ∀ σ ∈ L ω ( G ) , ∀ σ (cid:48) (cid:22) σ , ∃ σ (cid:48)(cid:48) ∈ T ∗ : σ (cid:48) σ (cid:48)(cid:48) (cid:22) σ ∧ | (cid:96) ( σ (cid:48)(cid:48) ) | ≤ K ⇒ |C ( w ) | = 1 , where w = (cid:96) ( σ (cid:48) σ (cid:48)(cid:48) ) . (cid:5) Therefore, an LPN system is periodically strongly detectableif the current and the subsequent states of the system can beperiodically determined for all trajectories of the system.
Definition 2.9: [ Periodically weak detectability ] An LPNsystem G = ( N, M , E, (cid:96) ) is periodically weakly detectable if ∃ K ∈ N , ∃ σ ∈ L ω ( G ) , ∀ σ (cid:48) (cid:22) σ , ∃ σ (cid:48)(cid:48) ∈ T ∗ : σ (cid:48) σ (cid:48)(cid:48) (cid:22) σ ∧ | (cid:96) ( σ (cid:48)(cid:48) ) | ≤ K ⇒ |C ( w ) | = 1 , where w = (cid:96) ( σ (cid:48) σ (cid:48)(cid:48) ) . (cid:5) In words, an LPN system is periodically weakly detectableif we can periodically determine the current state of the systemfor some trajectories of the system.
Example 2.10:
Let us consider again the LPN system inFig. 1. Its RG is shown in Fig. 2(a), and the observer ofRG is shown in Fig. 2(b). When ( ac ) ∗ is observed (the LPNsystem fires ( t t t ) ∗ ), the current state of the system can be uniquely determined, that is M . However, there alwaysexists two arbitrarily long prefix ( t t t ) ∗ t and ( t t t ) ∗ t (they have the same observation ( ac ) ∗ a ) such that the currentstate cannot be determined, that is, if ( ac ) ∗ a is observed, thecurrent state could be any state in { M , M , M } . Therefore,according to Definitions 2.6, the LPN system is not stronglydetectable.On the other hand, when the LPN system fires ( t t t ) ∗ andwe observe ( ac ) ∗ , we know that the current state of the systemis M periodically (after seeing c ). And when observing ( b ) ∗ (the LPN system fires ( t ) ∗ ), M is the current state of thesystem. Therefore, according to Definitions 2.7, 2.8 and 2.9,the LPN system is weakly detectable, periodically stronglydetectable and periodically weakly detectable. (cid:5) III. RG
AND ITS DETECTOR
In automation framework, detector was proposed to verify-ing the strong detectability and periodically strong detectabil-ity [10], [12], [20]. As [10] shows that the complexity ofconstruction of the detector is polynomial with respect tothe number of states of the system, which is lower than theobserver. Obviously, the same approach can be used on thebounded LPN system. Thus, in this section, we construct thedetector of the RG of the bounded LPN system, to check thestrong detectability and periodically strong detectability of theLPN system.As in [10], the detector of the RG is denoted by D = ( Q, E, f r , q ) , where Q ⊆ R ( N,M ) is a finite set of states. Since it isassumed that the marking from which the observation is gen-erated is not known, the initial state of D is q = R ( N, M ) ,and the other states of D is q ⊆ R ( N, M ) ∧| q | ≤ . The eventset of the detector is the alphabet E . We denote as U R ( M ) = { M (cid:48) ∈ N m | M [ σ u (cid:105) M (cid:48) , σ u ∈ T ∗ u } the unobservable reach ofthe marking M . The transition function f r : Q × E → Q isdefined as follows.Given a state q ⊆ R ( N, M ) , e ∈ E , t ∈ T, (cid:96) ( t ) = e . Let q t = U R ( { M ∈ R ( N, M ) |∃ M (cid:48) ∈ q, M (cid:48) [ t (cid:105) M } ) , then, f r ( q, e ) = { q t } if | q t | = 1 ; { q (cid:48) | q (cid:48) ⊆ q t ∧ | q (cid:48) | = 2 } if | q t | ≥ ; undef ined otherwise. Example 3.1:
Consider again the LPN system in Fig. 1, itsRG is shown in Fig. 2(a). By the construction method, thedetector of the BRG is presented in Fig. 3. The initial state isall the markings of the RG in Fig. 2(a). When a is observedat the initial state, there are four markings may be reachedin the RG. Thus according to the construction method, theinitial state can reach six states with a combination of thefour markings. (cid:5) Essentially, the detector of RG is constructed by splittingand recombining the state in C ( w ) when C ( w ) contains morethan two elements. Namely, for any state q = f r ( M , w ) in D , q ⊆ C ( w ) .Since detectability considers the transition sequences ofinfinite length, we first study the properties of cycles in thedetector of the RG. t (b) p t ( c ) t ( b )p t ( c ) t (a)p p p p t ( a ) t ( a ) t ( b ) p p a p p cp p ba a bb c p p ,p a p ,p p ,p cp p ,p b a a b b cp t (b)p p t ( a ) t ( a )t (a) p p bp a a a X p ,p a p ,p p ,p a a p p b b ba a a at (a) p p t ( ε )t ( a ) p p t ( b )t ( ε ) t ( c )p p M =p +p M =p M =p +p M =p M =p +p t ( ε ) t ( a ) t (a) t ( b )t ( ε ) t ( c ) (M ,M ) (M ,M ,M ) (M )(M ) a ac b b all {M }b {M ,M ,M } {M }ac bc a b all {M } b {M ,M } {M }a c b c ab{M ,M } {M ,M }a a bc a aall {(M ,0)}b {(M ,0),(M ,1)}{(M ,0)} a c bc a b c t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p t ( a ) t ( a )t (a) t ( b )t ( ε )t ( c ) (M ) (M ,M ,M ) (M ) (M ) a ac b(M ) aall {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)}a c bcb {(M ,0),(M ,1)} c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0)} {(M ,0)}aa bc b c a bbcall {M } b {M ,M ,M ,M }{M }a c bc b {M ,M ,M }c a ba all {M ,M } {M ,M } {M ,M }{M ,M }{M ,M }{M ,M } {M }{M } a a a aa a aa aa aa bbb b ccc c b b{(M ,0)} {(M ,0)}{(M ,1)}{(M ,0)} t ( a ) t ( a ) t (a) t ( b ) t ( c ) t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p p t ( ε ) M =p t (a) t ( ε )t ( a ) t ( b ) t ( a ) t ( c ) t ( ε )all {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)} a c bcb {(M ,0),(M ,1)}c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)}{(M ,0),(M ,1)} {(M ,0)}{(M ,0)}aa bc b c a b bc all {M }b {M ,M ,M ,M }{M } a c bc b {M ,M ,M }c a ba all {M ,M }{M ,M }{M ,M }{M ,M }{M ,M } {M ,M } {M } {M } a a aaaa a a aa a a bbb b c c c c b (M ,0) (M ,0) (M ,1) (M ,1) t ( a ) t ( a )t (a) t ( b ) t ( c ) Fig. 3. The detector of the RG in Fig. 2(a).
Definition 3.2: A (simple) cycle in the detector D =( Q, E, f r , q ) of a RG is a path γ j = q j e j q j . . . q jk e jk q j that starts and ends at the same state but without repeatededges, where q ji ∈ Q and e ji ∈ E with i = { , , . . . , k } ,and ∀ m, n ∈ { , , . . . , k } with m (cid:54) = n , q jm (cid:54) = q jn . Thecorresponding observation of the cycle is w = e j . . . e jk . Astate q ji contained in γ j is denoted by q ji ∈ γ j . (cid:5) Since RG is actually an automation, thus, we can concludethe following theorem according to [10].
Theorem 3.3:
Let G = ( N, M , E, (cid:96) ) be an LPN systemwhose T u -induced subnet is acyclic, and D = ( Q, E, f r , q ) the detector of its RG. The LPN system G is stronglydetectable iff for any q ∈ Q reachable from a cycle in D ,it is | q | = 1 .In words, an LPN system is strongly detectable if and onlyif in the detector of the RG, such that all the states reachablefrom any cycle that the cardinality of these states is 1. Theorem 3.4:
Let G = ( N, M , E, (cid:96) ) be an LPN systemwhose T u -induced subnet is acyclic, and D = ( Q, E, f r , q ) the detector of its RG. The LPN system G is periodicallystrongly detectable iff for any cycle γ j in D , ∃ q ∈ γ j , | q | = 1 .In words, an LPN system is periodically strongly detectableif and only if in the detector of the BRG, such that all thecycles have a state whose cardinality is 1. Remark 1 : Although the construction of the detector ac-cording to [10] is polynomial time complexity, it is knownthat the complexity of finding all the cycles in a directedgraph is NP-complete. Thus, the complexity of the detectorbased approaches proposed in [10] is not actually polynomialtime. However, finding all the strongly connected components(SCC) is of polynomial complexity w.r.t the size of the graph.Clearly, if a state of the observer is reachable from a cycle, itis also reachable from an SCC. Therefore, Theorem 3.3 canbe rephrased as follows.
Corollary 3.5:
Let G = ( N, M , E, (cid:96) ) be an LPN systemwhose T u -induced subnet is acyclic, and D = ( Q, E, f r , q ) the detector of its RG. The LPN system G is stronglydetectable iff for any q ∈ Q reachable from an SCC in D ,it is | q | = 1 . Remark 2 : According to Theorem 3.4, we also need tocheck all the cycles and we cannot take advantage from theusage of SCCs. However, it is easy to find that we can checkTheorem 3.4 by its contrapositive. Thus we just need to findone cycle according to the following corollary, which makesthe approach polynomial complexity.
Corollary 3.6:
Let G = ( N, M , E, (cid:96) ) be an LPN systemwhose T u -induced subnet is acyclic, and D = ( Q, E, f r , q ) the detector of its RG. The LPN system G is not periodicallystrongly detectable iff there exists a cycle γ j in D , for anystates q ∈ γ j , | q | (cid:54) = 1 . Example 3.7:
Consider again the LPN system in Fig. 1.Its RG is shown in Fig. 2(a), and the detector of the RG isshown in Fig. 3. Now we use Theorem 3.3 and 3.4 to checkits strong detectability and periodically strong detectability.In the detector of RG, we can see that there is a cycle γ = { M } a { M , M } c { M } containing state { M , M } whose cardinality is 2, thus, there exists a cycle that doesnot satisfy all states | q | = 1 . Therefore, the LPN system is notstrongly detectable.On the other hand, the state { M } in γ satisfy | q | = 1 . Andwe cannot find a cycle that all its states is | q | (cid:54) = 1 . Therefore,according to Corollary 3.6, the LPN system is periodicallystrongly detectable. (cid:5) IV. BRG
AND ITS DETECTOR
Checking detectability properties based on Theorem 3.3to 3.4 (or Corollary 3.5 to 3.6) requires the construction ofa RG and its detector. It is known that, the complexity ofconstructing the RG of a Petri net system is exponential inthe size of the net (number of places, transitions, tokens in theinitial marking). Therefore, to verify the detectability of largedimension systems, such an approach may not be feasible.In our previous work [4], [5], we show how the abovefour detectability properties can be verified using the notionof basis marking and observer, thus avoiding an exhaustiveenumeration of all the states in the RG. In this way, the stateexplosion problem is practically avoided [24]. However, thestep of building the observer is exponential complexity in theworst case.Since the BRG is usually much smaller than the RG andthe complexity of constructing the detector is lower than theobserver, thus, in this subsection, we build the BRG of theLPN system and explore the detector of the BRG to verifyingstrong detectability and periodically strong detectability.
A. BRG
Using the notion of basis marking, we introduce the basisreachability graph (BRG) for detectability. To guarantee thatthe BRG is finite, we assume that the LPN system is bounded.For each basis marking M b ∈ M b a binary scalar is assignedby function Ψ( M b ) : M b → { , } that are defined byEqs. (3): Ψ( M b ) = if M b + C u · y u ≥ has apositive integer solution; otherwise. (3) t (b) p t ( c ) t ( b )p t ( c ) t (a)p p p p t ( a ) t ( a ) t ( b ) p p a p p cp p ba a bb c p p ,p a p ,p p ,p cp p ,p b a a b b cp t (b)p p t ( a ) t ( a )t (a) p p bp a a a X p ,p a p ,p p ,p a a p p b b ba a a at (a) p p t ( ε )t ( a ) p p t ( b )t ( ε ) t ( c )p p M =p +p M =p M =p +p M =p M =p +p t ( ε ) t ( a ) t (a) t ( b )t ( ε ) t ( c ) (M ,M ) (M ,M ,M ) (M )(M ) a ac b b all {M }b {M ,M ,M } {M }ac bc a b all {M } b {M ,M } {M }a c b c ab{M ,M } {M ,M }a a bc a aall {(M ,0)}b {(M ,0),(M ,1)}{(M ,0)} a c bc a b c t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p t ( a ) t ( a )t (a) t ( b )t ( ε )t ( c ) (M ) (M ,M ,M ) (M ) (M ) a ac b(M ) aall {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)}a c bcb {(M ,0),(M ,1)} c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0)} {(M ,0)}aa bc b c a bbcall {M } b {M ,M ,M ,M }{M }a c bc b {M ,M ,M }c a ba all {M ,M } {M ,M } {M ,M }{M ,M }{M ,M }{M ,M } {M }{M } a a a aa a aa aa aa bbb b ccc c b b{(M ,0)} {(M ,0)}{(M ,1)}{(M ,0)} t ( a ) t ( a ) t (a) t ( b ) t ( c ) t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p p t ( ε ) M =p t (a) t ( ε )t ( a ) t ( b ) t ( a ) t ( c ) t ( ε )all {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)} a c bcb {(M ,0),(M ,1)}c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)}{(M ,0),(M ,1)} {(M ,0)}{(M ,0)}aa bc b c a b bc all {M }b {M ,M ,M ,M }{M } a c bc b {M ,M ,M }c a ba all {M ,M }{M ,M }{M ,M }{M ,M }{M ,M } {M ,M } {M } {M } a a aaaa a a aa a a bbb b c c c c b (M ,0) (M ,0) (M ,1) (M ,1) t ( a ) t ( a )t (a) t ( b ) t ( c ) Fig. 4. The BRG of the LPN system in Fig. 1.
We denote as B = ( X, E, f, x ) the BRG for detectabilityof an LPN system G = ( N, M , E, (cid:96) ) , where X ∈ M b ×{ , } is a finite set of states, and each state x ∈ X of the BRGis a pair ( M b , Ψ( M b )) . We denote as x (1) , x (2) the firstand the second element of x respectively. The initial stateof the BRG is x = ( M , Ψ( M )) . The event set of theBRG is identical to the alphabet E . The transition relation f : X × E → X can be determined by the following rule. Ifat marking M b there is an observable transition t for whicha minimal explanation exists and the firing of t and one ofits minimal explanations leads to M (cid:48) b , then an edge fromnode ( M b , Ψ( M b )) to node ( M (cid:48) b , Ψ( M (cid:48) b )) labeled with (cid:96) ( t ) is defined in the BRG. The procedure to construct the BRGfor detectability is summarized in Algorithm 1 in [4]. Example 4.1:
Let us consider again the LPN system in Fig. 1whose T u -induced subnet is acyclic. The LPN system has 6reachable markings and only 4 of them are basis markings,namely, M , M , M , M . When M b in Eq. (3) equals M , theequation has one positive integer solution. Thus, Ψ( M ) = 1 .On the other hand, for M , Eq. (3) does not have a positivesolution. Therefore, Ψ( M ) = 0 . The same for other basismarkings, thus, according to Algorithm 1 in [4], the BRG fordetectability is the graph in Fig. 4. Note that in Fig. 4 noinitial state is pointed out since the initial state is assumed tobe unknown. (cid:5) Lemma 4.2: [4] Let G be an LPN system whose T u -inducedsubnet is acyclic, M b ∈ M b a basis marking of G . If Ψ( M b ) =1 , there exists an observation w such that |C ( w ) | (cid:54) = 1 .In a simple word, if Ψ( M b ) = 1 , then there exists an ob-servation w such that |C ( w ) | contains more than one marking.However, even if Ψ( M b ) = 0 there may be another basismarking M (cid:48) b such that M b , M (cid:48) b ∈ C ( w ) . In this case, |C ( w ) | isstill not equal to 1.In the following, we construct the detector of the BRG toderive necessary and sufficient conditions for detectability. B. detector of the BRG
We construct a detector of the BRG B = ( X, E, f, x ) fordetectability as in [10]: B d = ( Q d , E, f d , q d ) , where Q d ⊆ X is a finite set of states. The initial state of B d is q d = X , and the other states of B d is q d ⊆ X ∩ | q d | ≤ .The event set of the detector is the alphabet E . The transitionfunction f d : Q d × E → Q d is defined as follows. t (b) p t ( c ) t ( b )p t ( c ) t (a)p p p p t ( a ) t ( a ) t ( b ) p p a p p cp p ba a bb c p p ,p a p ,p p ,p cp p ,p b a a b b cp t (b)p p t ( a ) t ( a )t (a) p p bp a a a X p ,p a p ,p p ,p a a p p b b ba a a at (a) p p t ( ε )t ( a ) p p t ( b )t ( ε ) t ( c )p p M =p +p M =p M =p +p M =p M =p +p t ( ε ) t ( a ) t (a) t ( b )t ( ε ) t ( c ) (M ,M ) (M ,M ,M ) (M )(M ) a ac b b all {M }b {M ,M ,M } {M }ac bc a b all {M } b {M ,M } {M }a c b c ab{M ,M } {M ,M }a a bc a aall {(M ,0)}b {(M ,0),(M ,1)}{(M ,0)} a c bc a b c t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p t ( a ) t ( a )t (a) t ( b )t ( ε )t ( c ) (M ) (M ,M ,M ) (M ) (M ) a ac b(M ) aall {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)}a c bcb {(M ,0),(M ,1)} c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0),(M ,1)} {(M ,0)} {(M ,0)}aa bc b c a bbcall {M } b {M ,M ,M ,M }{M }a c bc b {M ,M ,M }c a ba all {M ,M } {M ,M } {M ,M }{M ,M }{M ,M }{M ,M } {M }{M } a a a aa a aa aa aa bbb b ccc c b b{(M ,0)} {(M ,0)}{(M ,1)}{(M ,0)} t ( a ) t ( a ) t (a) t ( b ) t ( c ) t (a) p p t ( ε )t ( a )p p t ( b )t ( a ) t ( c ) p p M =p +p M =p M =p +p M =p M =p +p p t ( ε ) M =p t (a) t ( ε )t ( a ) t ( b ) t ( a ) t ( c ) t ( ε )all {(M ,0)}b {(M ,0),(M ,0),(M ,1)}{(M ,0)} a c bcb {(M ,0),(M ,1)}c a b a all a aa {(M ,0),(M ,1)} {(M ,0),(M ,1)}{(M ,0),(M ,1)} {(M ,0)}{(M ,0)}aa bc b c a b bc all {M }b {M ,M ,M ,M }{M } a c bc b {M ,M ,M }c a ba all {M ,M }{M ,M }{M ,M }{M ,M }{M ,M } {M ,M } {M } {M } a a aaaa a a aa a a bbb b c c c c b (M ,0) (M ,0) (M ,1) (M ,1) t ( a ) t ( a )t (a) t ( b ) t ( c ) Fig. 5. The detector of the BRG in Fig. 4.
Given a state q d ⊆ X, e ∈ E , let q t = { x ∈ X |∃ x (cid:48) ∈ q d , x ∈ f ( x (cid:48) , e ) } , then, f d ( q d , e ) = { q t } if | q t | = 1 ; { q (cid:48) d | q (cid:48) d ⊆ q t ∧ | q (cid:48) d | = 2 } if | q t | ≥ ; undef ined otherwise. Example 4.3:
Consider again the LPN system in Fig. 1, itsBRG is shown in Fig. 4. By the construction method, the de-tector of the BRG is presented in Fig. 5. The initial state is allthe basis markings of the BRG in Fig. 4. When a is observed atthe initial state, there are three basis markings may be reachedin the BRG. Thus according to the construction method, theinitial state can reach three states with a combination of thethree basis markings. (cid:5) Essentially, the detector of BRG is constructed by splittingand recombining the state in C b ( w ) . When |C b ( w ) | > , thedetector pairs all states in C b ( w ) in groups of tow. Namely,for any state q d = f d ( M , w ) in B d , (cid:83) x ∈ q d x (1) ⊆ C b ( w ) ⊆C ( w ) . Lemma 4.4:
Let G be an LPN system whose T u -inducedsubnet is acyclic, and B d = ( Q d , E, f d , q d ) the detector ofits BRG. If there exists a state q d ∈ Q d such that | q d | = 2 ,then there exists an observation w ∈ E such that |C ( w ) | (cid:54) = 1 . Proof:
Since by assumption | q d | = 2 , let q d = { x , x } , x (cid:54) = x . According to the construction of the detector ofBRG, then there must exists an observation w such that f d ( M , w ) = q d = { x , x } , x (cid:54) = x . Thus x (1) , x (2) ∈C ( w ) . Therefore, |C ( w ) | (cid:54) = 1 .In a simple word, if | q d | = 2 , then there exists an observa-tion w such that C ( w ) contains more than one marking. Lemma 4.5:
Let G be an LPN system whose T u -inducedsubnet is acyclic, and B d = ( Q d , E, f d , q d ) the detector ofits BRG. if there exists a state q d ∈ Q d such that ∃ x ∈ q d that x (2) = 1 , then there exists an observation w ∈ E such that |C ( w ) | (cid:54) = 1 . Proof:
Follow from Lemma 4.2.In a simple word, if ∃ x ∈ q d that x (2) = 1 , then thereexists an observation w such that C ( w ) contains more thanone marking. Proposition 4.6:
Let G be an LPN system whose T u -inducedsubnet is acyclic, and B d = ( Q d , E, f d , q d ) the detector of itsBRG. There exists an observation w ∈ E such that |C ( w ) | (cid:54) = 1 ,iff there exists a state q d ∈ Q d such that | q d | = 2 or ∃ x ∈ q d that x (2) = 1 . Proof: (If) Follow from Lemma 4.4 and Lemma 4.5.(Only if) Assume that there exists an observation w ∈ E such that |C ( w ) | (cid:54) = 1 , thus there exists two different markings M , M ∈ C ( w ) with M (cid:54) = M . According to the construc-tion of the detector of BRG, if M , M ∈ C b ( w ) , thus theremust exist a state q d ∈ Q d such that | q d | = 2 ; if M , M notall in C b ( w ) , thus there must exist a state q d ∈ Q d such that ∃ x ∈ q d that x (2) = 1 .In words, in an LPN system, there exists an observation w such that C ( w ) contains more than one marking, if and onlyif there exists a state q d such that | q d | = 2 or ∃ x ∈ q d that x (2) = 1 . Corollary 4.7:
Let G be an LPN system whose T u -inducedsubnet is acyclic, and B d = ( Q d , E, f d , q d ) the detector ofits BRG. If ∀ q d ∈ Q d , | q d | = 2 or ∃ x ∈ q d that x (2) = 1 . theLPN system G does not satisfy any detectability property. Proposition 4.8:
Let G be an LPN system whose T u -inducedsubnet is acyclic, and B d = ( Q d , E, f d , q d ) the detector of itsBRG. If there exists an observation w ∈ E such that |C ( w ) | =1 , then there exists a state q d ∈ Q d such that q d = { ( M b , } ,where M b ∈ M b . Proof:
Since by assumption |C ( w ) | = 1 , let C ( w ) = { M b } , according to the construction of the detector of BRG,then (cid:83) x ∈ q d x (1) = C ( w ) = { M b } . Thus, | (cid:83) x ∈ q d x (1) | = 1 ,i.e, there is only one state in q d . Since |C ( w ) | = 1 , byLemma 4.2, x (2) = 0 . Therefore, q d = { ( M b , } .In words, if there exists an observation w such that C ( w ) contains only one marking, then the corresponding state q d in B d contains only one basis marking M b and Ψ( M b ) = 0 .However, the converse is not true.Similar to Section III, we denote the simple cycles in thedetector of the BRG as follows:A (simple) cycle in the detector B d = ( Q d , E, f d , q d ) of aBRG is a path τ j = q j e j q j . . . q jk e jk q j that starts and endsat the same state but without repeated edges, where q ji ∈ Q d and e ji ∈ E . The corresponding observation of the cycle is w = e j . . . e jk . Theorem 4.9:
Let G be an LPN system whose T u -inducedsubnet is acyclic, and B d = ( Q d , E, f d , q d ) the detector ofits BRG. The LPN system G is strongly detectable iff for any q d ∈ Q d reachable from a cycle in B d , it is q d = { ( M b , } ,where M b ∈ M b . Proof:
Please see Appendix A for the proof.In words, an LPN system is strongly detectable if and onlyif in the detector of the BRG, such that all the states reachablefrom any cycle have the form { ( M b , } , i.e., there is only oneelement ( M b , Ψ( M b )) in these states and Ψ( M b ) = 0 .According to Remark 1, we can also take advantage fromthe usage of SCCs. Thus, Theorem 4.9 can be rephrased asfollows. Corollary 4.10:
Let G be an LPN system whose T u -inducedsubnet is acyclic, and B d = ( Q d , E, f d , q d ) the detector ofits BRG. The LPN system G is strongly detectable iff for any q d ∈ Q d reachable from an SCC in B d , it is q d = { ( M b , } ,where M b ∈ M b . Theorem 4.11:
Let G be an LPN system whose T u -inducedsubnet is acyclic, and B d = ( Q d , E, f d , q d ) the detector of itsBRG. The LPN system G is strongly periodically detectableiff for any cycle τ j in B d , ∃ q d ∈ τ j , q d = { ( M b , } , where M b ∈ M b . Proof:
Please see Appendix B for the proof.
In words, an LPN system is periodically strongly detectableif and only if in the detector of the BRG, such that all thecycles have a state having the form { ( M b , } .By Remark 2, Theorem 4.11 can also be written as follows. Corollary 4.12:
Let G be an LPN system whose T u -inducedsubnet is acyclic, and B d = ( Q d , E, f d , q d ) the detectorof its BRG. The LPN system G is not strongly periodicallydetectable iff there exists one cycle τ j in B d , for all q d ∈ τ j , q d (cid:54) = { ( M b , } , where M b ∈ M b . Example 4.13:
Consider again the LPN system in Fig. 1.Its BRG is shown in Fig. 4, and the detector of the BRGis shown in Fig. 5. Now we use Theorem 4.9 and 4.11to check its strong detectability and periodically strong de-tectability. In Fig. 5, we can see that there is a cycle τ = { ( M , } a { ( M , , ( M , } c { ( M , } containingstate { ( M , , ( M , } whose cardinality is 2 and Ψ( M ) =1 , thus, there exists a cycle that does not satisfy all states q d = { ( M b , Ψ( M b )) } with Ψ( M b ) = 0 . Therefore, the LPNsystem is not strongly detectable.On the other hand, the state { ( M , } in τ satisfy the form { ( M b , } . And in another cycle τ = { ( M , } c { ( M , } ,the only state { ( M , } also satisfy the form { ( M b , } ,therefore, the LPN system is periodically strongly detectable. (cid:5) V. C
ONCLUSION AND FUTURE WORK
In this paper, a novel approach to verifying detectability ofbounded labeled Petri nets is developed. Our approach is basedon the basis marking, and on the exploration of its detectorfor the detectability. For Petri nets whose unobservable subnetis acyclic, the strong detectability and periodically strongdetectability property can be decided by just constructingthe detector of the BRG. Since a complete enumeration ofpossible firing transition sequences is avoided and there is noneed for the construction of observer, the proposed approachis of lower complexity than the previous approaches. Thefuture research is to study on an algorithm that can checkthe weak detectability and periodically weak detectability withlow complexity. A
CKNOWLEDGMENT
This work was supported by the National Natural ScienceFoundation of China under Grant No. 61803317, the Fun-damental Research Funds for the Central Universities underGrant No. 2682018CX24, the Sichuan Provincial S&T Inno-vation Project under Grant No. 2018027, and the Key programfor International S&T Cooperation of Sichuan Province underGrant No. 2019YFH0097.R
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A. Proof of Theorem 4.9 (If) Assume LPN system G is not strongly detectable, that isfor all K ∈ N , there exist σ ∈ L ω ( G ) and σ (cid:48) (cid:22) σ, | w | ≥ K ⇒|C ( w ) | (cid:54) = 1 , where w (cid:48) = (cid:96) ( σ (cid:48) ) . Since σ is of an infinite lengthand B d has a finite number of nodes, the path along (cid:96) ( σ ) = w must contain a cycle τ j = q j e j q j . . . q jk e jk q j , i.e., thereexist w , w ∈ E ∗ , such that w = w ( e j . . . e jk ) ∗ w where | w | is finite. Since the T u -induced subnet is acyclic, let aprefix σ (cid:48) of σ , (cid:96) ( σ (cid:48) ) = w w (cid:48)(cid:48) , | (cid:96) ( σ (cid:48) ) | ≥ K , where w (cid:48)(cid:48) (cid:22) ( e j . . . e jk ) ∗ w . Let q d = f d ( q d , w w (cid:48)(cid:48) ) , since |C ( w ) | (cid:54) = 1 , w (cid:48) = (cid:96) ( σ (cid:48) ) = w w (cid:48)(cid:48) , by proposition 4.6, thus | q d | = 2 or ∃ x ∈ q d that x (2) = 1 . Namely, there exists a state q d reachablefrom a cycle in B d , it is | q d | = 2 or ∃ x ∈ q d that x (2) = 1 .(Only if) Assume there exists a cycle τ j in B d , q r ∈ τ j , w (cid:48) ∈ E ∗ , such that q d = f d ( q r , w (cid:48) ) is defined and | q d | = 2 or ∃ x ∈ q d that x (2) = 1 . Clearly, B d hasa finite number of nodes, then there exists an observation w such that q r = f d ( q d , w ) , with | w | is finite. Since q r ∈ τ j , then there exists σ ∈ L ω ( G ) and w = (cid:96) ( σ ) , w , w ∈ E ∗ such that w = w ( e j . . . e jk ) ∗ w . Since the T u -induced subnet is acyclic, then there exist σ (cid:48) (cid:22) σ with (cid:96) ( σ (cid:48) ) = w w (cid:48)(cid:48) , | (cid:96) ( σ (cid:48) ) | ≥ K , where w (cid:48)(cid:48) (cid:22) ( e j . . . e jk ) ∗ w and f d ( q d , w ) = q r . By assumption q d = f d ( q r , w (cid:48) ) isdefined and | q d | = 2 or ∃ x ∈ q d that x (2) = 1 , and f d ( q d , w w (cid:48) ) = f d ( q r , w (cid:48) ) = q d , thus by Proposition 4.6,this implies that the implication |C ( w w (cid:48) ) | (cid:54) = 1 holds. B. Proof of Theorem 4.11 (If) Assume LPN system G is not periodically stronglydetectable, that is for all K ∈ N , there exist σ ∈ L ω ( G ) and σ (cid:48) (cid:22) σ , ∀ σ (cid:48)(cid:48) ∈ T ∗ , (cid:96) ( σ (cid:48) σ (cid:48)(cid:48) ) = w (cid:48) : σ (cid:48) σ (cid:48)(cid:48) (cid:22) σ, | (cid:96) ( σ (cid:48)(cid:48) ) | ≤ K ⇒ |C ( w (cid:48) ) | (cid:54) = 1 . Since σ is of an infinite length and B d has a finite number of nodes, the path along (cid:96) ( σ ) = w must contain a cycle τ j = q j e j q j . . . q jk e jk q j , i.e., thereexist w ∈ E ∗ such that w = w ( e j . . . e jk ) ∗ where | w | isfinite. Let (cid:96) ( σ (cid:48) ) = w , (cid:96) ( σ (cid:48)(cid:48) ) = w (cid:48)(cid:48) (cid:22) ( e j . . . e jk ) ∗ . Since | (cid:96) ( σ (cid:48)(cid:48) ) | ≤ K , any state f d ( q d , w (cid:48) ) = f d ( q d , w w (cid:48)(cid:48) ) = q jr must in the cycle τ j . Since |C ( w (cid:48) ) | (cid:54) = 1 , by Proposition 4.6,it is | q jr | = 2 or ∃ x ∈ q jr that x (2) = 1 .(Only if) Assume there exists cycle τ j = q j e j q j . . . q jk e jk q j in B d , ∀ q jr ∈ τ j , | q jr | = 2 or ∃ x ∈ q jr that x (2) =1 . Clearly, there exist σ ∈ L ω ( G ) and w ∈ E ∗ , such that (cid:96) ( σ ) = w = w ( e j e j . . . e jk ) ∗ with | w | is finite. Since the T u -induced subnet is acyclic, then there exists σ with (cid:96) ( σ ) = w , for all σ ∈ T ∗ with (cid:96) ( σ ) = w (cid:22) ( e j e j . . . e jk ) ∗ , f d ( q d , w w ) = q jr ∈ τ j . By assumption | q jr | = 2 or ∃ x ∈ q jr that x (2) = 1 , therefore, by Proposition 4.6, |C ( w w ) | (cid:54) =1=1