On Deadlock Analysis and Characterization of Labeled Petri Nets with Undistinguishable and Unobservable Transitions

1. Introduction

Nowadays, the use of artificial dynamical systems is remarkably increasing in view of the enormous evolution of computer technology. The most important characteristic of these systems is determined by the occurrence of asynchronous events such that they are naturally and soundly labeled as discrete-event systems (DESs). In fact, many contemporary cyber-physical systems can be investigated from the view of discrete-event systems, where the procedure of system analysis and synthesis greatly depends on system mathematical models [1]. Without such a mathematical model, it would be impossible to analyze and control the behavior of a system. Different models are used to achieve this goal such as Petri nets, automata, Markov chains, etc. However, Petri nets have been recognized as one of the most powerful tools for modeling dynamic discrete-event systems. The increasing interest in Petri nets is stimulated by their modeling power and a mathematical arsenal supporting the analysis of the modeled systems through the usage of linear algebra and engineering optimization techniques. Nevertheless, several fundamental problems remain open. Among them, we in this research center around the deadlock problem, which is an important issue in designing automated systems [2] and has received much attention from the community of computer science and systems control engineering [3,4,5].

Researchers and practitioners have developed an array of deadlock resolution methods such as the early work by Zhang and Holloway [6], where a controlled Petri net model is used for a forbidden state avoidance issue under partial event observation and the initial marking of the net model is assume to be known. In [7,8] Li and Wonham seminally explore the use of state-feedback control of discrete-event systems under partial state observation, where a plant is modeled with a finite state automaton in which an event can be controllable (observable) or uncontrollable (unobservable). Their goal is to find necessary and sufficient conditions for the existence of “optimal” state feedback control laws given a control specification represented by a finite state automaton. Proposed in [13] is a deadlock characterization and a control procedure based on an observer of a plant, which is actually what an external agent can observe, representing the system output from the lens of outside observation. By virtue of the characterization, a systematic procedure is presented for deadlock recovery, i.e., a deadlock state can be guided/recovered to a predefined normal state. The main contribution is that the observer, the controller, and the deadlock recovery algorithm are all based on the same linear algebraic techniques. Later, Giua and Seatzu [10], based on the previously mentioned characterization, use generalized mutual exclusion constraints (GMECs) to ensure the control objective, where a plant is modeled with a Petri net and a GMEC serves as the control specification. They explore a recovery procedure and present a deadlock recovery algorithm using linear algebraic techniques. In these studies, contrary to the supervisory control theory in the Ramadge-Wonham framework [45], all transitions are in general assumed to be controllable and observable, and ordinary Petri nets are usually used to model systems. Partially to this end, but not limited to, the methodologies of deadlock analysis and control derived from Petri nets are not as general as those from the Ramadge-Wonham paradigm.

In the literature, people have followed three main lines to deal with the problem of deadlocks in DESs modeled with Petri nets. First, we mention the deadlock prevention principle [19,21,22,23,24,25] which is an offline computation procedure that imposes restrictions on system evolution to prevent the reachability of dead markings. The prevention is ensured by forbidding all first-met bad markings (FBMs) that represent the first entry from the live zone (LZ) to the dead zone (DZ) that are the partition of the reachability graph of a plant, where the initial state that is assumed to be legal belongs to the live zone.

Contrary to deadlock prevention principle, deadlock avoidance [26,27,28,29,30,31] is an on-line mechanism used to handle the blocking problem in Petri nets. As the mechanism proceeds online, at each state, a proper decision is made and issued to the actuators among all possible evolutions, aiming to avoid deadlocks and keep the system never trapped in a deadlock state. However, similar to the previously mentioned method, deadlock avoidance uses the notions of LZ and DZ to achieve this purpose. Although it generally leads to higher resource utilization and throughput, overly aggressive policies are still not able to totally eliminate all deadlocks in some cases. From here comes the necessity to use deadlock recovery strategies. Note that a recovery scheme is usually used in the scenario where deadlocks do not cause serious consequences. To summarize, conservative methods eliminate all unsafe states and deadlocks, and often some good states, thereby degrading the system performance. On the other hand, they are intended to be easy to implement. Finally, we briefly explain deadlock detection and recovery techniques [32,33,34,35]. The difference between this mechanism and the aforementioned two methodologies is that a deadlock detection and recovery approach permits the occurrence of a deadlock. Once it occurs, it is detected and then a recovery policy is initialized to guide and navigate the system to a predefined deadlock-free state, by simply reallocating the resources held by the system processes. The efficiency and efficacy of this approach depend upon the response time of the implemented algorithms for deadlock detection and recovery. In general, these algorithms require a large amount of data and may become complex when several types of shared resources are considered [36].

To summarize, the deadlock problem has often been discussed in the literature and several original theoretical approaches have been proposed [9,10,11,12,13] to solve it. While the previous works usually use ordinary Petri nets to model a system where all transitions are observable (the occurrence of any transition can be detected) and distinguishable (different transitions carry different labels), we deal, in this paper, with the issue of deadlock analysis and control in labeled Petri nets where transitions can be unobservable and undistinguishable since labeled Petri nets are an important framework of discrete-event systems and their liveness is usually a prerequisite for many problems such as fault diagnosis, opacity verification and enforcement, and state estimation [14]. We are motivated by the importance of defining deadlock and controlling a discrete-event system to ensure the correct and safe functioning of large complex systems. More preciously, the importance of defining the deadlock problem optimally will lead consequently to finding an optimal solution to guarantee the deadlock-freeness of the studied system. Moreover, given the lack of work treating this problem in labeled Petri nets, our contributions clearly appear.

This paper will be divided into seven main sections. Section 2 provides a background on Petri nets and gives the necessary notations. Section 3 introduces a study of basis reachability graphs. In Section 4, we give reachability and basis reachability graph analysis for the deadlock problem. An algorithm is developed to define an observed graph as a compact representation of the reachability graph of a Petri net with respect to the existence of dead reaches in Section 5. Experiment results are utilized to illustrate the proposed method in Section 6. Finally, Section 7 concludes the paper, along with future topics in the direction.

2. Preliminaries

2.1. Automata

A non-deterministic finite-state automaton (NFA) is a quadruple $\mathcal{A}=(X,E,\Delta ,x_0)$, where X is the finite set of states, $E=\{a,b,\dots ,c\}$ is the alphabet (usually finite), $\Delta \subseteq X\times E_{\epsilon }\times X$ is the transition relation with $E_{\epsilon }=E\cup \left\{\epsilon \right\}$, where $\epsilon$ is the empty string describing unobservable events, and $x_0\in X$ is the initial state. The transition relation specifies the dynamics of the NFA: if $(x,e,x^{\prime })\in \Delta$, then from state x the occurrence of event $e\in E_{\epsilon }$ yields the state $x^{\prime }$. In particular, given any state $x\in X$, we have $(x,\epsilon ,x)\in \Delta$. The transition relation can be extended to ${\Delta }^{*}\subseteq X\times E^{*}\times X$, where $(x_0,\omega ,x_k)\in {\Delta }^{*}$ if there exists a sequence of events and states $x_0{e}_{j1}{x}_1\dots x_{k-1}{e}_{jk}{x}_k$ such that $\sigma =e_{j1}\dots e_{jk}$ generates the word $\omega \in E^{*}$, $x_{ij}\in X$ for $i=1,\dots ,k$, and $e_{ji}\in E_{\epsilon }$, $(x_{ji-1},e_{ji},x_{ji})\in \Delta$ for $i=1,2,\dots ,k$.

An event e is said to be defined at state $x_i$ if there exists a state $x_j\in X$ such that $(x_i,e,x_j)\in \Delta$. An NFA is simply denoted as $\mathcal{A}=(X,E,\Delta )$ in the case where the initial state could be any state in X or the initial state is of no interest. The generated language of an automaton $\mathcal{A}=(X,E,\Delta )$ from a state $x\in X$ is defined as

$$\mathcal{L}(\mathcal{A},x)=\{\omega \in E^{*}\mid \exists x^{\prime }\in X:(x,\omega ,x^{\prime })\in {\Delta }^{*}\}.$$

(1)

Given a set of states $Y\subseteq X$, we define the language generated from the states in set Y as:

$$\mathcal{L}(\mathcal{A},Y)=\bigcup _{x\in Y}\mathcal{L}(\mathcal{A},x).$$

(2)

2.2. Petri Nets

In this section, we review basic definitions of Petri nets. For more details we refer readers to [15] and [17].

Definition 1.

(Petri net). A Petri net is a quadruple $N=(P,T,\mathrm{Pre},\mathrm{Post})$, where P is a finite and non-empty set of places; T is a finite and non-empty set of transitions; $\mathrm{Pre}:P\times T\to \mathbb{N}$ and $\mathrm{Post}:P\times T\to \mathbb{N}$ are respectively the pre- and post- incidence functions that specify the arcs directed from places to transitions, and vice versa 1. We use $\mathcal{C}=\mathrm{Post}-\mathrm{Pre}$ to represent the incidence matrix of a Petri net. The cardinality of the set of places is denoted as $m=\left|P\right|$ and that of the set of transitions is $n=\left|T\right|$.

The input and output sets of a node $x\in P\cup T$ are defined by ^•$x=\{y\mid (y,x)\in F\}$ and $x^{•}=\{y\mid (x,y)\in F\}$, respectively, with $F\subseteq (P\times T)\cup (T\times P)$ is the set of all directed arcs from transitions to places or from places to transitions, specified by the pre- and post incidence functions. This notation can be extended to a set of nodes as follows: given $X\subseteq P\cup T$, one has ^•$X={{\bigcup }_{x\in X}}^{•}x$ and $X^{•}={\bigcup }_{x\in X}{x}^{•}$. A marking of a net is a mapping $M:P\to \mathbb{N}$ that assigns to each place a non-negative integer number of tokens. We denote by $M\left(p\right)$ the marking of a place p at the marking M. For computational convenience, a marking can be represented by a vector of size m. A Petri net system $\langle N,M_0\rangle$ is a net N with initial marking $M_0$. Graphically, places are represented by circles and transitions by bars while black dots are used to represent tokens that flow among the places via transition firing. As to be seen, using the enabling and firing rules, the dynamics of a Petri net is defined. In fact, their role is to determine the flow of tokens between places, thus specifying how the initial marking can evolve.

Definition 2.

(Firing Rule). Given a Petri net system $\langle N,M_0\rangle$, a transition $t\in T$ is enabled at a marking M if $M\ge Pre(·,t)$. An enabled transition can fire, yielding a new marking $M^{\prime }$ such that $M^{\prime }=M+\mathcal{C}(·,t)$. We write $M_0[\sigma \rangle M$ to denote that the sequence of transitions $\sigma =t_{j_1}\cdots t_{j_k}$ is enabled at $M_0$ and $M[\sigma \rangle M^{\prime }$ to denote that the firing of σ yields $M^{\prime }$. Given a sequence $\sigma \in T^{*}$, the function $\pi :T^{*}\to {\mathbb{N}}^n$ associates with a transition sequence σ in $T^{*}$ the Parikh vector $y_{\sigma }=\pi \left(\sigma \right)\in {\mathbb{N}}^n$, i.e., $y\left(t\right)=k$ if transition t appears k times in σ.

Definition 3.

(Reachable Marking). A marking M is reachable in $\langle N,M_0\rangle$ if there exists a sequence σ such that $M_0[\sigma \rangle M$. The set of all markings reachable from $M_0$ defines the reachability set of $\langle N,M_0\rangle$, denoted by $R(N,M_0)$, i.e., $R(N,M_0)=\{M\in {\mathbb{N}}^{\left|P\right|}\mid \exists \sigma \in T^{*}:M_o[\sigma \rangle M\}$.

A Petri net system is bounded if there exists a non-negative integer $k\in \mathbb{N}$ such that for any place $p\in P$ and for any reachable marking $M\in R(N,M_0)$, $M\left(p\right)\le k$ holds. A marking M is said to be a deadlock or dead marking if no transition is enabled at M. A Petri net is said to be deadlock-free if at least one transition is enabled at every reachable marking. A transition t is said to be live at $M_0$ if for any $M\in R(N,M_0)$, there exists a feasible sequence of transitions at M such that $M_0[\sigma \rangle M^{\prime }$ and t is enabled at $M^{\prime }$. A Petri net is said to be live at a marking if all transitions are live at the marking.

Definition 4.

(Induced Subnet). Given a net $N=(P,T,Pre,Post)$ and a subset $T^{\prime }\subseteq T$ of its transitions, the $T^{\prime }$–induced subnet of N is a net $N^{\prime }=(P,T^{\prime },Pr{e}^{\prime },Pos{t}^{\prime })$, where $Pr{e}^{\prime }$ (resp., $Pos{t}^{\prime }$) is the restriction of $Pre$ (resp., $Post$) to $T^{\prime }$.

The net $N^{\prime }$ can be thought of as a net obtained from N by removing all transitions in $T\setminus T^{\prime }$, which is also written as $N^{\prime }{\prec }_{T^{\prime }}N$. An essential notion in Petri net structural analysis is siphon [18]. A nonempty set $\mathcal{S}\subseteq P$ is a siphon if ^•$\mathcal{S}\subseteq {\mathcal{S}}^{•}$. A siphon is said to be minimal if there is no siphon contained in it as a proper subset. Siphons are a major structural technique for deadlock control in a Petri net system.

3. Labeled Petri Nets

For more details about labeled Petri nets, we refer the readers to [16]. A labeled Petri net (LPN) is a quadruple $G=(N,M_0,E,\ell )$, where

-

E is the alphabet, i.e., the set of labels;

-

$\langle N,M_0\rangle$ is a Petri net system;

-

$\ell :T\to E\cup \left\{ϵ\right\}$ is the labeling function that assigns to a transition $t\in T$ either a symbol from E or the empty string $\epsilon$.

As usual, $T_{uo}$ is used to denote the set of silent (unobservable) transitions, i.e., the set of transitions labeled with the empty string $\epsilon$. Hence, founded on this type of events, the set of transitions can be partitioned into two disjoint sets $T=T_o\cup T_{uo}$, where $T_o=\{t\in T\mid \ell \left(t\right)\in E\}$ is the set of observable transitions and $T_{uo}=\{t\in T\mid \ell \left(t\right)=\epsilon \}$ is called the set of unobservable transitions. By extending the labeling function to firing sequences $\ell :T^{*}\to E^{*}$, we define $\ell \left(\epsilon \right)=\epsilon$ and

$$\ell =\left\{\begin{array}{c}\ell \left(\sigma \right)=\epsilon \mathrm{if}\sigma \in T_{uo},\hfill \\ \ell \left(\sigma \right)=\sigma \mathrm{if}\sigma \in T_o,\hfill \\ \ell \left(\sigma t\right)=\ell \left(\sigma \right)\ell \left(t\right)\mathrm{with}\sigma \in T^{*}\mathrm{and}t\in T.\hfill \end{array}\right.$$

(3)

An example of an LPN is shown in Figure 1, where $P=\{p_1,p_2,p_3,p_4,p_5,p_6\}$ is the set of places and $T=\{t_1,t_2,t_3,t_4\}$ is the set of transitions with $\Sigma =\left\{c\right\}$, $T_o=\{t_2,t_3\}$, $T_o=\{t_1,t_4\}$, $\ell \left(t_1\right)=\ell \left(t_4\right)=\epsilon$, and $\ell \left(t_2\right)=\ell \left(t_3\right)=c$. Note that we will refer to this example in Section 4.

Given an LPN $G=(N,M_0,E,\ell )$ and a marking $M\in R(N,M_0)$, the language generated by G from M is defined as

$$\mathcal{L}(G,M)=\{\omega \in E^{*}\mid \exists \sigma \in T^{*}:M[\sigma \rangle \mathrm{and}\ell \left(\sigma \right)=\omega \}.$$

(4)

Furthermore, given a set of markings $Y\subseteq R(N,M_0)$ of G, we define the language generated from the markings in Y as

$$\mathcal{L}(G,Y)=\bigcup _{M\in Y}\mathcal{L}(G,M).$$

(5)

A string belonging to $\mathcal{L}(G,M_0)$ is called an observation. Let $\omega$ be an observation of an LPN $G=\langle N,M_0,E,\ell \rangle$. We define

$$\mathcal{C}\left(\omega \right)=\{M\in {\mathbb{N}}^m\mid \exists \sigma \in T^{*}:M_0[\sigma \rangle M\mathrm{and}\ell \left(\sigma \right)=\omega \}.$$

(6)

as the set of markings consistent with $\omega$. We also define the set of markings generating $\omega$ as

$$\mathcal{I}\left(\omega \right)=\{M\in R(N,M_0)\mid \exists \sigma \in T^{*},\exists M^{\prime }\in R(N,M_0):M[\sigma \rangle M^{\prime }\mathrm{and}\ell \left(\sigma \right)=\omega \}.$$

(7)

By definition, once the word (string) $\omega$ is observed, the sets $\mathcal{C}\left(\omega \right)$ and $\mathcal{I}\left(\omega \right)$ are non-empty sets. Given an observation $\omega \in \mathcal{L}(G,M_0)$, ${\omega }^{\prime }$ is a prefix of $\omega$, if and only if there exists a string $\nu \in E^{*}$ such that $\omega ={\omega }^{\prime }\nu$. We write $\overline{\omega }$ to denote the prefix set of $\omega$ with $\epsilon \in \overline{\omega }$ and $\omega \in \overline{\omega }$. For a set of observations $L\subseteq \mathcal{L}(G,M_0)$, we define the prefix-closure of L as $\overline{L}=\{\nu \in \mathcal{L}(G,M_0)\mid \exists \omega \in L:\nu \in \overline{\omega }\}$. In plain words, $\overline{L}$ consists of all the prefixes of all the strings in L. It is trivial that $\epsilon \in \overline{L}$ and for any string $\omega \in L$, $\omega \in \overline{L}$ if L is not empty.

4. Minimal Explanation and Basis Reachability Graph

The concept of basis reachability graph is proposed in [39,40], which serves as a compact way to represent the reachability set of an LPN to solve fault diagnosis and many other problems in the context of discrete-event systems. It has been used to address the issues related to observability, e.g., diagnosability and opacity verification and enforcement problems [39,40,41]. A generalization of the basis reachability graph is proposed by Ma and his colleagues in [38], where a more fundamental structure that only contains basis markings independently from the system’s partition is defined. In this section, we will provide some basic definitions that will be used in the following.

4.1. Minimal Explanation and Minimal E-Vector

Definition 5.

(Basis Partition [38]). Given a Petri net $N=(P,T,Pre,Post)$, a pair $\pi =(T_E,T_I)$ is called a basis partition of T if it satisfies:

$$\left\{\begin{array}{c}{T}_I\subseteq T,\hfill \\ T_E=T\setminus T_I,\hfill \\ T_I\text{–}\mathit{induced}\mathit{subnet}\mathit{is}\mathit{acyclic}.\hfill \end{array}\right.$$

(8)

In a basis partition $(T_E,T_I)$, the sets $T_E$ and $T_I$ are called the explicit transition set and the implicit transition set, respectively. That is, a basis partition is a partition of T into $T_E$ and $T_I$ such that the $T_I$–induced subnet is acyclic. We denote $|T_E|=n_E$ and $|T_I|=n_I$.

Given a Petri net $N=(P,T,Pre,Post)$, a basis partition $\pi =(T_E,T_I)$, a marking M, and a transition $t\in T_E$, we define the set of explanations of transition t at marking M as

$$\Sigma (M,t)=\{\sigma \in T_I^{*}\mid M[\sigma \rangle M^{\prime }\mathrm{and}{M}^{\prime }\ge Pre(·,t)\}.$$

(9)

In plain words, the firing of any implicit sequence $\sigma \in \Sigma (M,t)$ leads the system to a new marking at which the explicit transition t is enabled. Moreover, the set of e-vectors (explanation vectors), which represents the set of firing vectors associated with the firing sequences in $\Sigma (M,t)$ is defined as

$$Y(M,t)=\{y_{\sigma }\in {\mathbb{N}}^{n_I}\mid \sigma \in \Sigma (M,t)\}.$$

(10)

We also define the set of minimal explanations of transition t at marking M as

$${\Sigma }_{\min }(M,t)=\{\sigma \in \Sigma (M,t)\mid \nexists {\sigma }^{\prime }\in \Sigma (M,t):y_{{\sigma }^{\prime }}⪇y_{\sigma }\}.$$

(11)

The corresponding set of minimal explanation vectors is

$$Y_{\min }(M,t)=\{y_{\sigma }\in {\mathbb{N}}^{n_I}\mid \sigma \in {\Sigma }_{\min }(M,t)\}$$

(12)

Given a marking M and an explicit transition t, the work in [39,42] proposes an efficient algorithm to compute the set of minimal e-vectors $Y_{\min }(M,t)$.

4.2. Basis Reachability Graph

Definition 6.

(Basis Marking). Given a Petri net $N=(P,T,\mathrm{Pre},\mathrm{Post})$ with an initial marking $M_0$ and a basis partition $\pi =(T_E,T_I)$, its basis marking set $\mathcal{M}(N,M_0,\pi )$ is defined as follows:

• $M_0\in \mathcal{M}(N,M_0,\pi )$;
• If $M\in \mathcal{M}(N,M_0,\pi )$, then the predicate holds: $\forall y\in Y_{\min }(M,t):(M^{\prime }=M+{\mathcal{C}}_I·y+\mathcal{C}(·,t))\Rightarrow (M^{\prime }\in \mathcal{M}(N,M_0,\pi ))$

where ${\mathcal{C}}_I$ is the incidence matrix of N restricted to the transition set $T_I$. A marking M in $\mathcal{M}(N,M_0,\pi )$ is called a basis marking of $\langle N,M_0\rangle$ with respect to $\pi =(T_E,T_I)$.

Definition 7.

(Basis Reachability Graph). Given a bounded net $N=(P,T,\mathrm{Pre},\mathrm{Post})$ with an initial marking $M_0$ and a basis partition $\pi =(T_E,T_I)$, its basis reachability graph (BRG) is a non-deterministic finite state automaton, represented by a quadruple $\mathcal{B}=(\mathcal{M},T_r,\Delta ,M_0)$ such that

• the state set $\mathcal{M}$ is the set of basis markings;
• the event set $T_r$ is the set of pairs $(t,y)\in T_E\times {\mathbb{N}}^{n_I}$;
• the transition relation Δ is:

  $$\Delta =\{(M_1,(t,y),M_2)\mid t\in T_E,y\in Y_{\min }(M_1,t),\mathrm{and}{M}_2=M_1+\mathcal{C}·y+\mathcal{C}(·,t)\}$$
• the initial state is the initial marking $M_0$.

Algorithm 1 iteratively computes basis markings and constructs a BRG. We use $\mathcal{B}(N,M_0,\pi )$ to denote the BRG of a net $\langle N,M_0\rangle$ with basis partition $\pi$. If no confusion arises, it is simply written as $\mathcal{B}$.

By instructions (4–13), this algorithm computes the set of all minimal explanations of each selected marking M and associates each explanation $y\in Y_{\min }(M,t)$ with a new marking $M^{\prime }$ defined by the state equation of a basis marking $M^{\prime }=M+{\mathcal{C}}_I·y+\mathcal{C}(·,t)$. Each new marking is defined as an unchecked marking and this procedure runs iteratively until there are no unchecked markings. For a bounded Petri net, based on $\mathcal{M}\subseteq R(N,M_0)$, Algorithm 1 terminates in a finite time.

5. Deadlock Analysis

In this section, we will give a brief overview of reachability graph-based methodologies dealing with the deadlock problem in a DES modeled with Petri nets. In fact, when treating the problem, we can distinguish between structure and reachability graph analysis. In this paper, we are interested in the latter, providing effective solutions for the deadlock problem for an LPN.

5.1. Analysis of Reachability Graph

In [43], Uzam and Zhou propose to split a reachability graph of a net model into two parts: a deadlock zone (DZ) and a live zone (LZ). The former may contain deadlock states (markings), partial deadlock states, and states that inevitably lead a system to deadlocks or livelocks. A live zone constitutes the remaining good states of the reachability graph representing the optimal system behavior. We show in Figure 2 the partition of the reachability graph of some LPN. By a slight abuse of notation, we use $DZ$ and $LZ$ to denote the sets of markings in the deadlock zone and live zone, respectively.

The set ${\mathcal{M}}_L$ of legal markings is defined as ${\mathcal{M}}_L=\{M\mid M\in R(N,M_0)\mathrm{and}{M}_0\in R(N,M)\}$. An MTSI (marking/transition separation instance) is a pair of a marking M and a transition t, denoted as $(M,t)$, whose set is defined as $\mathsf{\Omega }=\{(M,t)\mid M[t\rangle M^{\prime },M\in LZ,\mathrm{and}{M}^{\prime }\in DZ\}$, where M is called a dangerous marking. The set of good markings is defined as ${\mathcal{M}}_G={\mathcal{M}}_L-{\mathcal{M}}_D$, where ${\mathcal{M}}_D=\{M\in R(N,M_0)\mid \exists t\in T:(M,t)\in \mathsf{\Omega }\}$ defines the set of dangerous markings, i.e., the set of deadlock states (markings) and the states which inevitably lead to deadlocks. Hence, by removing all dangerous markings from ${\mathcal{M}}_L$, we obtain the set of good markings. In the following, we recall some definitions originally from [20].

The transitions in a Petri net model are classified into two parts: critical and good ones, whose sets are denoted as $T_c$ and $T_g$, respectively, where $T_c=\{t\in T\mid \exists M\in R(N,M_0):(M,t)\in \mathsf{\Omega }\}$ and $T_g=\{t\in T\mid \nexists M\in R(N,M_0):(M,t)\notin \mathsf{\Omega }\}$. From a reachability graph, we can see that the liveness can be guaranteed by implementing all MTSIs.

For a transition t, a legal marking M is called a t-good marking if t is not enabled at M, or $M^{\prime }$ with $M[t\rangle M^{\prime }$ being a legal marking; M is called a t-dangerous marking if $M^{\prime }$ with $M[t\rangle M^{\prime }$ is an illegal marking. The sets of t-good and t-dangerous markings are denoted as ${\mathcal{G}}_t$ and ${\mathcal{D}}_t$, respectively. Let t be a transition and M a t-good marking. M is called a t-enabled good marking if $M[t\rangle$; otherwise, M is called a t-disabled good marking. The sets of t-enabled (resp., t-disabled) good markings is denoted by ${\epsilon }_t$ (resp., ${\epsilon }_{\overline{t}}$). Thus, ${\mathcal{G}}_t$ and ${\mathcal{M}}_L$ can be defined as ${\mathcal{G}}_t={\epsilon }_t\cup {\epsilon }_{\overline{t}}$ and ${\mathcal{M}}_L={\mathcal{G}}_t\cup {\mathcal{D}}_t$. Let t be a critical transition and M a t-dangerous marking. $(M,t)$ is called a t-critical MTSI. ${\mathsf{\Omega }}_t=\{(M,t)\mid M\in {\mathcal{D}}_t\}$ defines the set of t-critical MTSIs [37].

5.2. Analysis of Basis Reachability Graph

Under the assumption that the $T_u$-induced subnet is acyclic, the study in [39,40] proposes the original BRG to solve the fault diagnosis problem. A generalization of BRG has been introduced later in [38]. The proposed graph well preserves the information on the reachability set, as well as on the evolution behavior of the Petri net, while its structure is usually much more compact than the conventional reachability graph and the state explosion problem is to some extent mitigated. In this subsection, we focus on analyzing deadlocks based on BRGs and studying their effectiveness in characterizing deadlocks in systems modeled by LPN.

Definition 8.

[38] Given a net $N=(P,T,\mathrm{Pre},\mathrm{Post})$, a basis partition $\pi =(T_E,T_I)$, and a basis marking $M_b\in \mathcal{M}$, the implicit reach of $M_b$, denoted by $R_I\left(M_b\right)$, is defined as

$$R_I\left(M_b\right)=\{M\in {\mathbb{N}}^m\mid \exists \sigma \in T_I^{*}:M=M_b+\mathcal{C}·y_{\sigma }\}.$$

(13)

By definition, the implicit reach of a basis marking $M_b$ is the set of markings reachable from $M_b$ by firing only implicit transitions.

Corollary 1.

[38] Given a Petri net $\langle N,M_0\rangle$ and its BRG $\mathcal{B}$, it holds

$$R(N,M_0)=\bigcup _{M_b\in \mathcal{M}}{R}_I\left(M_b\right).$$

(14)

As Corollary 1 shows, we can easily observe that the union of the implicit reaches of all basis markings equals the set of reachable markings. The main contribution of a BRG is that of preventing the exhaustive exploration of all reachable states. In fact, to construct a BRG, one only needs to explore the minimal e-vectors for each basis marking and observable transition. However, this may ignore useful information about deadlock markings. Let us now introduce the following definitions that are essential and critical to formalize and develop the main results in this paper.

Definition 9.

Given a marking M and an explicit transition $t\in T_E$, the set of entire explanations at M is defined as

$$\mathcal{E}\left(M\right)=\bigcup _{t\in T_E}\Sigma (M,t).$$

(15)

We also define the set of entire explanation vectors as

$$Y_T\left(M\right)=\bigcup _{t\in T_E}Y(M,t).$$

(16)

Similarly, we introduce the set of entire minimal explanations at M as

$${\mathcal{E}}_{\min }\left(M\right)=\bigcup _{t\in T_E}{\Sigma }_{\min }(M,t).$$

(17)

and its corresponding set of entire minimal e-vectors as

$$Y_{T_{\min }}\left(M\right)=\bigcup _{t\in T_E}{Y}_{\min }(M,t).$$

(18)

In other words, $\mathcal{E}\left(M\right)$ represents the set of all implicit sequences whose firing at the marking M enables an explicit transition $t\in T_E$, while ${\mathcal{E}}_{\min }\left(M\right)$ is the set of all minimal implicit sequences feasible at $M_b$, at which any explicit transition $t\in T_E$ is enabled. The number of arcs in a BRG, where $M_b$ is the input vertices, is equal to the cardinality of ${\mathcal{E}}_{\min }\left(M_b\right)$.

Lemma 1.

Given an LPN $G=(N,M_0,E,\ell )$, a basis partition $\pi =(T_E,T_I)$, and an implicit transition $t_E$, a basis marking $M_b$ is reachable from another basis marking if it satisfies

$$M_b=\sum _{i=1}^n{\alpha }_i{M}_{b_i}+{\mathcal{C}}_I·y_I+\mathcal{C}(·,t_E)$$

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with $y_1\in Y_{\min }(M_b,t_E)$, ${\sum }_{i=1}^n{\alpha }_i=1$ and ${\alpha }_i\in \{0,1\}$.

Proof. 

This result stems from the state equation of a basis marking. Since ${\alpha }_i\in \{0,1\}$ and ${\sum }_{i=1}^n{\alpha }_i=1$, we have one and only one ${\alpha }_i=1$ for some i. In other words, we have at least one basis marking $M_{b_i}$ such that

$$M^{\prime }=M_{b_i}+{\mathcal{C}}_I·y_I+\mathcal{C}(·,t_E).$$

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Based on the state equation of a basis marking, if $M\in \mathcal{M}(N,M_0,\pi )$, then for all $t\in T_E$ and for all $y\in Y_{\min }(M,t)$, we have

$$M^{\prime }=M+{\mathcal{C}}_I·y+\mathcal{C}(·,t)\Rightarrow M^{\prime }\in \mathcal{M}(N,M_0,\pi ).$$

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Thus, we conclude that $M^{\prime }$ is a reachable basis marking. □

As indicated by the lemma, a marking M is a reachable basis marking if there exists, at least, another reachable basis marking from which M can be reachable by firing a minimal explanation followed by its corresponding explicit transition $t_E$.

Definition 10.

Given an LPN $G=(N,M_0,E,\ell )$, a basis partition $\pi =(T_E,T_I)$, and an implicit transition $t_E\in T_E$, a marking M is a dead basis marking (DBM) if it satisfies the following linear system:

$$DBM\left(M_b\right):=\left\{\begin{array}{cccc}\hfill M_b& =\sum _{i=1}^n{\alpha }_i{M}_{b_i}+{\mathcal{C}}_I·y_I+\mathcal{C}(·,t_E)\hfill & \hfill & \left(\mathit{a}\right)\hfill \\ \hfill \mathcal{E}\left(M_b\right)& =\varnothing \hfill & \hfill & \left(\mathit{b}\right)\hfill \\ \hfill \sum _{i=1}^n{\alpha }_i& =1\hfill & \hfill & \left(\mathit{c}\right)\hfill \\ \hfill {\alpha }_i& \in \{0,1\}\hfill & \hfill & \left(\mathit{d}\right)\hfill \\ \hfill y_1& \in Y_{\min }(M_b,t_E)\hfill & \hfill & \left(\mathit{e}\right)\hfill \end{array}\right.$$

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In words, a marking M is said to be a DBM if there exists a basis marking $M_{b_i}$, from which it is reachable, by firing $\sigma ={\sigma }_I{t}_E$ with ${\sigma }_I$ a minimal explanation of $t_E$, and there does not exist any implicit sequence feasible at $M_b$ that can enable an explicit transition. Note that, despite the set of entire explanations at $M_b$ is empty, the set of implicit reaches $R_I\left(M_b\right)$ could not be. From Definition 10, we derive the following sufficient condition for deadlock existence in an LPN.

Theorem 1.

(Sufficient Condition) Given an LPN $G=(N,M_0,E,\ell )$, a basis partition $\pi =(T_E,T_I)$, a DBM $M_b$, and a marking $M^{\prime }\in {\mathbb{N}}^m$, the following predicate holds:

$$M^{\prime }\in R_I\left(M_b\right)\Rightarrow M^{\prime }\in DZ$$

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Proof. 

The fact that $M_b$ is a DBM implies $\mathcal{E}\left(M_b\right)=\varnothing$. We have $R(N,M_b)=R_I\left(M_b\right)$ and for all $M^{\prime }\in R_I\left(M_b\right)$ such that $M_0[\sigma \rangle M^{\prime }$, it holds $\sigma \notin \mathcal{E}\left(M_b\right)$, leading to the truth of $M^{\prime }\in DZ$. □

Proposition 1.

Given an LPN $G=(N,M_0,E,\ell )$, a basis partition $\pi =(T_E,T_I)$, and a marking M. If M is a BDM, then $M\in DZ$ holds.

Proof. 

The proof of Proposition 1 derives from Theorem 1. Suppose that M is a dead basis marking. Theorem 1 has shown $R(N,M)=R_I\left(M\right)\subseteq DZ$. We conclude that from M only the markings in $DZ$ are reachable. Hence, $M\in DZ$ holds. □

Theorem 1 gives a sufficient condition to make decision about the safeness of a basis marking’s reaches. In fact, once the set of entire explanations feasible at $M_b$ is empty, we can easily conclude that only implicit transitions can be feasible at $M_b$. This means that the reachability set of the basis marking $M_b$ is equal to its implicit reach. Note that a BRG is founded under the assumption of acyclicity of the $T_I$-induced subnet. Therefore, there does not exist any implicit sequence feasible at $M_b$ that can return the system to its optimal behavior. In this case, any marking $M^{\prime }$ reachable from $M_b$ ( with $M^{\prime }\in R_I\left(M_b\right)$) should be necessarily a dead state, a partial deadlock state, or an inevitable state that leads to deadlocks.

Example 1.

Let us consider the LPN in Figure 3, whose transition partition is $\pi =(T_I,T_E)$ with $T_E=\{t_1,t_4,t_5\}$ and $T_I=\{t_2,t_3\}$. The initial marking is $M_0=p_1+p_2$. As shown in Figure 4, the reachability graph contains seven markings, from which we can easily extract the deadlock zone $DZ=\{M_3,M_5\}$. The BRG of this example is visualized in Figure 5. The basis marking $M_{b2}$ is a DBM since it is reachable from $M_{b0}$ by firing $(t_5,y)$, where $y=[0,0]$ and $\mathcal{E}\left(M_{b2}\right)=\varnothing$. Consider the implicit reach of $M_{b2}$, $R_I\left(M_{b2}\right)=\left\{M_5\right\}\subseteq DZ$ holds. By Proposition 1, we have $M_{b2}\in DZ$.

Although DMBs are covered by a BRG, it is not able to fully characterize the deadlock problem in an LPN. In fact, the basis reachability graph covers only deadlock states reachable by firing minimal explanations, neglecting thus other feasible implicit events that may threaten the system and lead to deadlocks. To clearly explain this point, let us introduce Example 2 which illustrates the weakness of a BRG in deadlock recognization.

Example 2.

Consider the Petri net in Figure 1 with an initial marking $M_0$, where $M_0\left(p_1\right)=2$, $M_0\left(p_5\right)=M_0\left(p_6\right)=1$, and $M_0\left(p\right)=0$ for other places. The set of transitions T is divided into $T_0=\{t_2,t_3\}$ and $T_{uo}=\{t_1,t_4\}$.

The reachability graph of the net model, shown in Figure 6, has five reachable markings. The firing of $t_3$ from the marking $M_2$ leads the system to a dead marking $M_4$. While considering a basis partition $\pi =(T_E,T_I)$ with $T_E=T_0$ and $T_I=T_{uo}$, Figure 7 defines its BRG. As shown in the figure, this graph is composed of three basis markings. Regarding deadlocks, it gives $DZ=\left\{M_4\right\}$. However, since this dead state is not reachable by firing a minimal explanation, the BRG of this example does not contain any DBM and the sequence of transitions leading the system to the deadlock zone is not taken into account by the BRG. Thus, the weakness of a BRG facing this type of deadlock clearly appears.

In this section, we have shown that using the original BRG we cannot fully address the deadlock problem. Thus, in the next section, we will introduce new concepts that are able to deal with deadlock characterization and analysis in an LPN.

6. Dangerous Implicit Reaches

In this section, we propose new concepts that enable us to identify the existence of deadlocks in systems modeled by an LPN. A system under consideration in this research is supposed to be arbitrarily labeled, i.e., no condition or restriction is imposed on the labeling function $\ell :T^{*}\to E^{*}$. Hence, the system may contain undistinguishable transitions, i.e., at a marking M, two (or more) enabled transitions may share the same label (e.g., a). We denote the set of undistinguishable transitions with respect to a label $\sigma$ as $T_{ud}\left(\sigma \right)=\{t\in T|\ell \left(t\right)=\sigma \in E_{\epsilon }\}$. Moreover, the set of distinguishable transitions in an LPN is denoted by $T_d$. By definition, $T_{uo}\subseteq T_{ud}$ holds, where $T_{ud}={\bigcup }_{\sigma \in E_{\epsilon }}{T}_{ud}\left(\sigma \right)$ is the set of undistinguishable transitions. Hence, the set of transitions can be partitioned as follows $T=T_d\cup T_{udo}\cup T_{uo}$, where $T_{udo}$ is the set of undistinguishable transitions but observable, i.e., $T_{udo}={\bigcup }_{\sigma \in E_{\epsilon }}{T}_{udo}\left(\sigma \right)$, where $T_{udo}\left(\sigma \right)=\{t\in T|\ell \left(t\right)=\sigma \in E\}$.

Consider a system modeled with an LPN and consistent with the aforementioned description. With respect to the concept of basis reachability graph, we can classify the set of implicit reachable states from each basis marking $M_b\in \mathcal{M}$ to three main subsets: minimal explanations, non minimal explanations, and non explanation sequences. As we have proven in the previous section, if a deadlock state is reachable by firing an explicit transition preceded by one of its minimal explanations, the BRG can identify such a deadlock state. However, the BRG is still unable to identify deadlocks reachable by firing non minimal explanations or implicit sequences that can never belong to the set of entire explanations.

6.1. Dangerous Implicit Markings (DIMs) and Dangerous Implicit Vectors (DIVs)

At the beginning of this subsection, let us introduce some results that will be used in the following to minimize the related computations.

Theorem 2.

Suppose that $\sigma =t_1{t}_2\in T^{*}$ is firable starting from a marking $M_1$ with $M_1[t_1\rangle M_2[t_2\rangle M_3$ in an PLN. It holds:

$$\left(M_1[t_1\rangle M_2[t_2\rangle M_3\right)\wedge (\exists M_4\in {\mathbb{N}}^m:M_1[t_2\rangle M_4)\Rightarrow M_4[t_1\rangle M_3$$

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Proof. 

Let $\sigma =t_1{t}_2$ and $M_1[t_1\rangle M_2[t_2\rangle M_3$. This implies that $M_3=M_1+\mathcal{C}(·,t_1)+\mathcal{C}(·,t_2)\ge 0$. Suppose that $t_2$ is firable from $M_1$ with $M_1[t_2\rangle M_4$. Then, we have $M_4=M_1+\mathcal{C}(·,t_2)\ge 0$. Hence we get the following system

$$\left\{\begin{array}{c}{M}_1+\mathcal{C}(·,t_1)+\mathcal{C}(·,t_2)\ge 0\hfill \\ M_1+\mathcal{C}(·,t_2)\ge 0\hfill \end{array}\right.$$

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which indicates $M_4+\mathcal{C}(·,t_1)\ge 0$. Accordingly, we conclude that from $M_4$ we necessarily have $M_4[t_1\rangle$ and the reachable marking is $M_3$. □

Corollary 2.

Given two sequences $\sigma ,{\sigma }^{\prime }\in T^{*}$, where $\sigma =t_1{t}_2\cdots t_n$, $t_i\in \sigma$, and ${\sigma }^{\prime }=\sigma \setminus \left\{t_i\right\}$, suppose that there is a marking $M_1\in {\mathbb{N}}^m$ from which $t_1,t_2,\cdots ,t_n$ are firable sequentially with $M_1[t_1\rangle M_2[t_2\rangle \cdots [t_n\rangle M_n$. If ${\sigma }^{\prime }$ is firable from $M_1$ with $M_1[{\sigma }^{\prime }\rangle M_x$, then $M_x[t_i\rangle M_n$ holds.

Proof. 

The proof of this corollary derives from Theorem 2. In fact, suppose that from a marking $M\in {\mathbb{N}}^m$ we have $M[\sigma \rangle M^{\prime }$. This implies that (1) $M+C_y\ge 0$ with $y=\pi \left(\sigma \right)$ and ${\mathcal{C}}_y=\mathcal{C}(·,t_1)+\mathcal{C}(·,t_2)+\cdots +\mathcal{C}(·,t_n)$. If $M[{\sigma }^{\prime }\rangle M^{\prime \prime }$, then (2) $M^{\prime \prime }=M+\mathcal{C}(·,t_1)+\cdots +\mathcal{C}(·,t_{i-1})+\mathcal{C}(·,t_{i+1})+\cdots +\mathcal{C}(·,t_n)$ and $\mathcal{C}(·,t_{i+1})+\cdots +\mathcal{C}(·,t_n)\ge 0$. By (1) and (2) we conclude $M^{\prime \prime }+\mathcal{C}(·,t_i)\ge 0$ and $M^{\prime \prime }+\mathcal{C}(·,t_i)=M^{\prime }$. □

As a matter of fact, Corollary 2 is a generalization of Theorem 2. In what follows, we touch upon a primary notion called dangerous implicit marking which is pivotal to the development of this research.

Definition 11.

Given an LPN $G=(N,M_0,E,\ell )$, a basis marking $M_b$, and a vector $y_I\in {\mathbb{N}}^{n_I}$, let $M_b^{\prime }=M_b+C_I·y_I$ with $y_I\ge 0$ and $\nexists y_I^{\prime }\in Y_T\left(M_b\right)$ such that $y_I^{\prime }\ge y_I$. Then $y_I$ is said to be a dangerous implicit vector (DIV) while $M_b^{\prime }$ is called a dangerous implicit marking (DIM).

Thus, we define the set of Dangerous Implicit Vectors (DIVs) at $M_b$ by

$$Y_{\beta }\left(M_b\right)=\{y\in {\mathbb{N}}^{n_I}\mid M_b^{\prime }=M_b+{\mathcal{C}}_I·y_I\ge 0\mathrm{and}(\nexists y_I^{\prime }\in Y_T\left(M_b\right):y_I^{\prime }\ge y_I)\}$$

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Moreover, we define the set of Dangerous Implicit Markings (DIMs) reachable from $M_b$ as

$$M_{\beta }\left(M_b\right)=\{M^{\prime }\in {\mathbb{N}}^m\mid M_b^{\prime }=M_b+{\mathcal{C}}_I·y_I\ge 0\mathrm{and}{y}_I\in Y_{\beta }\left(M_b\right)\}$$

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In words, consider a basis marking $M_b$ and a vector $y\in {\mathbb{N}}^{n_I}$ feasible at $M_b$. The vector $y_I$ is dangerous because it can never be an explanation vector of any explicit transition at $M_b$, i.e., for any feasible implicit sequence of transition ${\sigma }^{\prime }$ at $M^{\prime }$, where $M^{\prime }=M_b+{\mathcal{C}}_I·y_I$ is the marking reachable by firing $y_I$ from $M_b$, the sequence ${\sigma }^{\prime }{\sigma }_I$ can never belong to the set of entire explanations at $M_b$. In this case, the marking reachable by firing an implicit sequence ${\sigma }_I$ from $M_b$, with $y_I=\pi \left({\sigma }_I\right)$, is called a dangerous implicit marking.

Theorem 3.

Given an LPN $G=(N,M_0,E,\ell )$, a basis marking $M_b$, and a sequence of implicit transitions ${\sigma }_I$ satisfying:

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let $S=\{M^{\prime }\in {\mathbb{N}}^m\mid M^{\prime }=M_b+{\mathcal{C}}_I·y_I^{\prime }\ge 0,{\sigma }_I^{\prime }\in T_I^{*},y_I^{\prime }=\pi \left({\sigma }_I^{\prime }\right),\mathrm{and}{\sigma }_I^{\prime }\in \overline{{\sigma }_I^{\prime }}\}$ be a set of markings, where $\overline{{\sigma }_I^{\prime }}$ denotes the prefix set of ${\sigma }_I^{\prime }$.3 Then, for any $M\in S$, $M\in DZ$.

Proof. 

Let $M_b\in {\mathbb{N}}^m$ be a basis marking and ${\sigma }_I,{\sigma }_I^{\prime }\in T_I^{*}$ be two implicit sequences with ${\sigma }_I\in \overline{{\sigma }_I^{\prime }}$. Suppose that $M_b$ and ${\sigma }_I$ verify Theorem 3. Since ${\sigma }_I\in \overline{{\sigma }_I^{\prime }}$ and ${\sigma }_I\notin \mathcal{E}\left(M_b\right)$, we have ${\sigma }_I^{\prime }\notin \mathcal{E}\left(M_b\right)$. Let us consider now $S=\{M^{\prime }\in {\mathbb{N}}^m\mid M^{\prime }=M_b+{\mathcal{C}}_I·y_I^{\prime }\ge 0\}$. We aim to prove that for all $M^{\prime }\in S$, $M^{\prime }\in DZ$.

(a)

By $M_b[{\sigma }_I\rangle$, $M^{\prime }=M_b+{\mathcal{C}}_I·y_I^{\prime }>0$. Moreover, there does not exist $t_E\in T_E$ such that $M^{\prime }[t_E\rangle$ due to ${\sigma }_I^{\prime }\notin \mathcal{E}\left(M_b\right)$.

(b)

As known, the $T_I$-induced subnet is acyclic, implying that the predicate holds: $\nexists {\sigma }_I=t_{I_1},t_{I_2},\dots ,t_{I_k}$: $M_1[t_{I_1}\rangle M_2[t_{I_2}\rangle \dots M_k[t_{I_k}\rangle M_1$. That is to say, from the marking $M^{\prime }$, we cannot co-reach $M_0$ by firing only implicit transitions.

By (a) and (b), we conclude that for all $M^{\prime }\in S$, $M_0\notin R(N,M^{\prime })$ holds. □

In simple words, consider an LPN $G=(N,M_0,E,\ell )$, an arbitrary basis partition $\pi =(T_E,T_I)$, and a basis marking $M_b$. Under the assumption that the $T_u$-induced subnet is acyclic, we have proven by Theorem 3 that any marking $M^{\prime }$ reached by firing an implicit sequence ${\sigma }_I$ that does not belong to $\overline{\mathcal{E}\left(M_b\right)}$, the prefix-closure of the set of entire explanations $\mathcal{E}\left(M_b\right)$, is either a dead marking or a marking that inevitably leads the system to a blocking state. Thus, $M^{\prime }$ belongs necessarily to the dead Zone (DZ).

Proposition 2.

Given an LPN $G=(N,M_0,E,\ell )$, a basis marking $M_b$, and a dangerous implicit vector $y_I\in Y_{\beta }\left(M_b\right)$ with $M_I^{\prime }=M_b+{\mathcal{C}}_I·y_I$, $M_b^{\prime }$ is a DIM and for all $y_I^{\prime \prime }\in {\mathbb{N}}^{n_I}$ such that $M_b^{\prime \prime }=M_b^{\prime }+{\mathcal{C}}_I·y_I^{\prime \prime }\ge 0$, $M_B^{\prime \prime }\in DZ$ holds.

Proof. 

Given a basis marking $M_b$ and a sequence of implicit transitions ${\sigma }_I\in T_I^{*}$ firable from $M_B$ with $\pi \left({\sigma }_I\right)=y_I$, suppose that $y_I$ is a DIV, i.e., there does not exist $y_I^{\prime }\in Y_T\left(M_b\right)$ such that $y_I^{\prime }\ge y_I$. This implies that for all $y_x\in {\mathbb{N}}^{n_I}$, there does not exist $y_I^{\prime }\in Y_T\left(M_b\right)$ such that $y_I^{\prime }\ge y_I+y_x$. We have for all $\sigma ={\sigma }_I{\sigma }_x$ such that $M_b[\sigma \rangle$, $\sigma \notin \overline{\mathcal{E}\left(M_b\right)}$ is true and thus by Theorem 3, $M_b^{\prime \prime }=M_b+{\mathcal{C}}_I·y_{\sigma }\in DZ$ holds. □

By virtue of DIV and DIM notions, Proposition 2 represents a generalization of Theorem 3. In fact, when checking the safeness of each non dead-basis marking, if an implicit vector $y_I$ is feasible at a basis marking $M_B$ ($M_b+{\mathcal{C}}_I·y_I\ge 0$) and there does not exist any explanation vector $y_I^{\prime }$ feasible at $M_b$ such that $y_I^{\prime }\ge y_I$, we do not need to check the maximal cardinality that the implicit vector $y_I$ can reach, since it is certain by Proposition 2 that we will never find a vector included in the set of entire e-vectors.

In the following, Algorithm 2 is introduced to check the safeness of each non-dead basis marking. Given a basis marking $M_b$ and its set of entire e-vectors $Y_T\left(M_b\right)$, the role of the proposed algorithm is to check if there exists any DIV, i.e., any implicit vector feasible at $M_b$ and does not belong to the set of entire explanations $\mathcal{E}\left(M_b\right)$, that can threaten the system deadlock-freedom, thereby providing more information about deadlocks that can be reached from the basis marking $M_b$.

Algorithm 2 works in a breadth-first manner. Considering the set of entire explanation vectors, this algorithm checks whether there exist any undesirable implicit vectors that can fire and lead to deadlocks. A first test is done by step 6 to verify the existence of the sufficient condition given by Theorem 3 and Proposition 2. Specifically, the matrix $\mathcal{C}$ defines the set of entire e-vectors. If there does not exist any row $j^{*}$ in C such that $\mathcal{C}(j^{*},·)\ge I_{n_{I}x{n}_I}(j,·)+B(i^{*},·)$, the conditions given by Theorem 3 and Proposition 2 are satisfied. Hence, the reached marking will be added to the set of dangerous implicit markings without adding it to the set ${\mathcal{M}}_{\mathrm{new}}$. A second condition is introduced by step 9 of Algorithm 2. Indeed, this test aims to minimize the computation by checking whether the conditions in Theorem 2 and Corollary 2 are satisfied. Once ${\mathcal{M}}_{\mathrm{new}}$ becomes empty, a set of dangerous implicit vectors DIVs and its corresponding set of implicit dangerous markings DIMs are obtained by $\Psi \left(M_b\right)=[{\mathcal{M}}_{\beta }\left(M_b\right)\mid Y_{\beta }\left(M_b\right)]$.

Proposition 3.

The complexity of Algorithm 2 is $O(2^{|T_I|}+\left[\right|T_I{|}^2·\left|P\right|+|T_I|·|Y_T\left(M_b\right)\left|\right])$.

Proof. 

In the worst case, the first step of the algorithm, i.e., the while loop, takes time of $O\left(2^{|T_I|}\right)$, where $|T_I|$ represents the number of implicit transitions of the studied Petri net. In the second step, browsing through the matrix ${\mathcal{C}}_I^T$ takes the time of $O\left(\right|T_I\left|\right)$. The third step considers the checking of the existence of the new implicit vector $I_{n_{I}x{n}_I}(j,·)+B(i^{*},·)$ in the two matrices B and $\mathcal{C}$ of size $O\left(\right|T_I\left|\right)$ and $O\left(\right|Y_T\left(M_b\right)\left|\right)$ respectively, with $|Y_T\left(M_b\right)|$ being the cardinality of the set of entire e-vectors of $M_b$. Therefore, the total cost for “the else if statement” is $O\left(\right|T_I|+|P|+|Y_T\left(M_b\right)\left|\right)$. Finally, the asymptotic total cost of Algorithm 2 is $O(2^{|T_I|}·\left[\right|T_I{|}^2+|T_I|·|P|+|T_I|·|Y_T\left(M_b\right)\left|\right])$. □

Although, in the worst case, the number of iterations of Algorithm 2 may grow exponentially with the diameter (i.e., length of the maximal path) of the implicit subnet, in most practical cases, this number is quite reasonable, and Algorithm 2 has good performance since each iteration mainly consists of simple additions or comparisons of vectors.

Example 3.

Let us consider the Petri net system shown in Figure 8, which consists of 9 places $P=\{p_1,\dots ,p_9\}$ and 8 transitions $T=\{t_1,\dots ,t_8\}$. The transition set partition is defined by $\pi =(T_E,T_I)$ with $T_E=T_o=\{t_1,t_4,t_5,t_6,t_8\}$ and $T_I=T_{uo}=\{t_2,t_3,t_7\}$. Consider the initial marking $M_0=p_1+p_7+p_9$. The reachability graph, shown in Figure 9, contains 16 reachable markings, 12 of which are legal markings and 4 of which are deadlock markings. In Figure 10, we present its BRG composed of 6 markings.

Let us focus on the basis marking $M_{b5}$ which is the marking $M_{14}$ in the reachability graph. As shown in the BRG, $M_{b5}$ is not a DBM. However, from the reachability graph one observes that $M_{15}$ is a dead marking reachable implicitly from $M_{14}$ and neglected by the BRG. By applying Algorithm 2, we can easily obtain $\Psi \left(M_b\right)=[{\mathcal{M}}_{\beta }\left(M_b\right)\mid Y_{\beta }\left(M_b\right)]$ with $Y_{\beta }\left(M_b\right)Y_{\beta }\left(M_b\right)=[0,1,0]$ and ${\mathcal{M}}_{\beta }\left(M_b\right)=\left[000001102\right]$. In fact, the set of entire e-vectors at $M_{14}$ contains only one element $Y_T\left(M_{b5}\right)=\left\{[0,0,1]\right\}$ which belongs to $\sum (M_{b5},a\left(t_1\right))$. However, the vector $y_T=[0,1,0]$ is feasible from $M_{b5}$. Since there does not exist any vector $y_T^{\prime }\in Y_T\left(M_{b5}\right)$ such that $y_T^{\prime }\ge y_T$, $y_T$ is a dangerous implicit vector at $M_{b5}$ and $M_d=M_{15}=\left[000001102\right]$ is its corresponding DIM. As Theorem 3 and Proposition 3 indicate, this DIM belongs to the dead zone DZ.

6.2. Non Minimal Explanations

As we have previously mentioned, the concept of dangerous implicit markings is able to deal with the deadlock states reachable by an implicit sequence that can never be an explanation of any explicit transition at a given basis marking. However, in this subsection, we touch upon the last type of dead markings reachable from a basis marking which is a non minimal explanation. In plain words, the principle of the basis reachability graph is to find the minimal implicit path to activate each explicit transition. By this principle, non minimal explanations will not be considered.

Proposition 4.

Given an LPN $G=(N,M_0,E,\ell )$, a basis partition $\pi =(T_E,T_I)$, a marking M, and a sequence ${\sigma }_I\in T_I^{*}$ such that $M[{\sigma }_I\rangle M^{\prime }$, if ${\sigma }_I\in \mathcal{E}\left(M\right)$, $M^{\prime }$ is not a dead marking.

Proof. 

Let $M\in {\mathbb{N}}^m$ be a marking and ${\sigma }_I\in T_I^{*}$ be a sequence of implicit transitions firable from M. ${\sigma }_I\in \sum (M,t)$ signifies that there exists a transition $t\in T_E$ such that $M[{\sigma }_I\rangle M^{\prime }[t\rangle$. We can then conclude that ${\sigma }_I$ does not lead to a dead marking. □

By definition, an explanation is a sequence of implicit transitions feasible at a given basis marking to enable an explicit transition. Thus, we conclude that a marking reachable from a basis one by firing an explanation of any explicit transition is not a dead marking.

Theorem 4.

Given an LPN $G=(N,M_0,E,\ell )$, a basis marking $M_b$, an explicit transition $t_E$, and $y_I\in Y(M_b,t_E)$ that is a non minimal explanation vector of $t_E$ at $M_b$, let $M\in {\mathbb{N}}^m$ be a marking reachable by firing $t_E$ preceded by ${\sigma }_I$, an explanation consistent with $y_I$. If M is a dead marking, then there exists a basis marking $M_b^{\prime }$ reachable from $M_b$ by firing ${\sigma }_{min}{t}_E$, where ${\sigma }_{min}$ is a minimal explanation of $t_E$ at $M_b$ in comparison with ${\sigma }_I$, from which a dangerous implicit marking M is reachable, i.e., $M_b[{\sigma }_{min}{t}_E\rangle M_b^{\prime }$, and $M\in {\mathcal{M}}_{\beta }\left(M_b^{\prime }\right)$.

Proof. 

Consider a basis marking $M_b$, an implicit transition $t_E\in T_E$, and a non minimal explanation vector $y_I\in Y(M_b,t_E)$, i.e., $y_I\notin Y_{min}(M_b,t_E)$. Suppose that $M=M_b+{\mathcal{C}}_I·y_I+\mathrm{Pre}(·,t_E)$ is a dead marking. The fact that $y_I$ is a non minimal e-vector implies that there exists an e-vector $y_I^{\prime }\in Y_{min}(M_b,t_E)$ verifying $y_I^{\prime }<y_I$. That is to say that there exists a vector $y_I^{\prime \prime }\in {\mathbb{N}}_I^n$ such that $y_I=y_I^{\prime }+y_I^{\prime \prime }$. Thus we have

$$\left\{\begin{array}{c}{M}_{b1}=M_b+{\mathcal{C}}_I·y_I^{\prime }+\mathcal{C}(·,t_E)\ge 0\hfill \\ M=M_b+{\mathcal{C}}_I·y_I^{\prime }+{\mathcal{C}}_I·y_I^{\prime \prime }+\mathcal{C}(·,t_E)\ge 0\hfill \end{array}\right.$$

We conclude that M can be reached from the basis marking $M_{b1}$ by firing an implicit sequence of transitions ${\sigma }_I\in T_I^{*}$ such that $y_I^{\prime }=\pi \left({\sigma }_I\right)$. However, M is a dead marking. Hence, there does not exist any e-vector $y\in Y_T\left(M_{b1}\right)$ such that $y\ge y_I^{\prime \prime }$. Finally, we obtain $M\in {\mathcal{M}}_{\beta }\left(M_{b1}\right)$. □

By Theorem 4, we have proved that if a dead state is reachable by firing a non minimal explanation, we can certainly reach it by firing only implicit transitions from a basis marking. Thus, the concept of dangerous implicit reach can cover this type of dead reachable states and identify dangerous sequences leading to such markings.

Example 4.

Consider again the LPN in Figure 8 whose unobservable subnet is acyclic. Let us focus our attention on the basis marking $M_{b1}$. The set of explanations of $t_4$ at $M_{b1}$ is $\sum (M_{b1},t_4)=\{t_2,t_2{t}_3,t_3{t}_2,t_2{t}_7,t_7{t}_2\}$. Therefore, we have ${\sum }_{min}(M_{b1},t_4)=\left\{t_2\right\}$ and $Y_{min}=\left\{[1,0,0]\right\}$. In the BRG, only the minimal explanation is considered. Although this minimal explanation leads the system to a legal state, we can observe that according to the reachability graph, the non minimal explanations $t_2{t}_7$ and $t_7{t}_2$ conduct the system inevitably to the dead state $M_{12}=\left[100010000\right]$. Since $M_{b1}[t_2{t}_4\rangle M_{b2}$ with ${\sigma }_I=t_2$ is the minimal explanation in comparison with the aforementioned two explanations, one sees $M_{12}\in {\mathcal{M}}_{\beta }\left(M_{b2}\right)$.

7. Observed Graph: A Deadlock Characterization for Labeled Petri Nets

In this section, we propose a new graph that is able to fully characterize deadlocks in systems modeled by labeled Petri nets.

Definition 12.

(Observed Graph) Given a bounded Petri net $N=(P,T,\mathrm{Pre},\mathrm{Post})$ with an initial marking $M_0$ and a basis partition $\pi =(T_E,T_I)$, the observed graph of the Petri net is a non-deterministic finite state automaton that is a quadruple $\mathcal{G}=({\mathcal{M}}_O,Tr,{\Delta }_O,M_0)$, where

• the state set ${\mathcal{M}}_O={\mathcal{M}}_B\cup {\mathcal{M}}_D$ is the set of observed markings with:

  -

  the state set ${\mathcal{M}}_B$ being the set of observed basis markings;

  -

  the state set ${\mathcal{M}}_D$ being the set of observed dead markings;

• the event set $Tr$ is the set of pairs $(t,y)\in T_E\cup \{\varnothing \}\times {\mathbb{N}}^{n_I}$;
• the transition relation is ${\Delta }_O={\Delta }_B\cup {\Delta }_D$ with:

  -

  ${\Delta }_B=\{(M_1,(t,y),M_2)\mid t\in T_E,y\in Y_{min}(M_1,t),\mathrm{and}{M}_2=M_1+{\mathcal{C}}_I·y+\mathcal{C}(·,t)\}$

  -

  ${\Delta }_D=\{(M_1,(\varnothing ,y),M_2)\mid y\in {\mathbb{N}}^n,y\in Y_{\beta }\left(M_1\right),\mathrm{and}{M}_2=M_1+{\mathcal{C}}_I·y\}$

• the initial state is the initial marking $M_0$.

Algorithm 3 presents a method to calculate the observed graph $\mathcal{G}$. By steps (3–16), we first compute the original BRG. By step 7, the set of entire explanation $Y_T\left(M_b\right)$ is computed. Steps (17–21) check the existence of the sufficient condition given by Theorem 1. If the tested marking M is a DBM, then we add it to the set of dead markings ${\mathcal{M}}_D$, assign a tag to M, and return to step 2 to check another marking without tag from $M_B$ if there exists. If the first sufficient condition of Theorem 1 is not satisfied, by steps (18–21) we check the safeness of the implicit reaches of the marking M. If $\beta$ (the set of dangerous implicit vectors and its corresponding markings computed by Algorithm 2) is not empty, for all $y_I\in \beta$ we add the reachable implicit marking $M_I$ to ${\mathcal{M}}_D$ and $(M,(\varnothing ,y_I),M_I)$ to ${\Delta }_D$. This procedure runs iteratively until there is no unchecked marking in ${\mathcal{M}}_B$. For a bounded Petri net, based on $({\mathcal{M}}_B\cup {\mathcal{M}}_D)\subseteq R(N,M_0)$, Algorithm 3 terminates in a finite time.

We first consider the complexity of the observed graph $\mathcal{G}$, i.e., the number of observed markings in it. In the worst case, when each transition in the net system has a self-loop, the partition $(T_E,T_I)$ is $T_E=T$ and $T_I=\varnothing$, the observed graph $\mathcal{G}$ has the same complexity as the basis reachability graph and the reachability graph, i.e., $|{\mathcal{M}}_B|=|{\mathcal{M}}_O|=|R(N,M_0)|$. Despite in the aforementioned case $\mathcal{G}$ has the same complexity as a reachability graph and BRG, we note that in practice we can usually find a basis partition $(T_E,T_I)$ in which $T_I\ne \varnothing$, under which $\mathcal{G}$ is much more compact than the reachability graph, i.e., $|{\mathcal{M}}_O|\ll |R(N,M_0)|$ and more precisely $|{\mathcal{M}}_B|\le |{\mathcal{M}}_O|\le |{\mathcal{M}}_B|+|\mathcal{D}|$, where $\left|\mathcal{D}\right|$ represents the number of dead markings.

Theorem 5.

Given an LPN $G=(N,M_0,E,\ell )$ and its observed graph $\mathcal{G}=({\mathcal{M}}_O,Tr,{\Delta }_O,M_0)$, if there is no dead marking in $\mathcal{G}$, G is a deadlock-free net.

Proof. 

This is trivial. Suppose that the observed graph of G does not contain any dead observed markings. There is no marking in G that can be a deadlock state at which no transition is enabled, which completes the proof. □

Example 5.

Consider again the Petri net system given by Figure 8 . The set of deadlock markings is $\mathcal{D}=\{M_{D1}=\left[000002001\right],M_{D2}=\left[100001000\right],M_{D3}=\left[100010000\right],M_{D4}=\left[000001102\right]\}$. In Figure 10, we present the BRG composed of 6 markings. However, only one dead marking $M_{b3}=\left[001001001\right]$ exists in the BRG.

Using Algorithm 3, we construct the observed graph of the Petri net system. The resulting graph is shown in Figure 11, which contains 8 markings. From $M_{b2}=\left[001010001\right]$, the set of DIVS and DIMs are $\beta =\Psi \left(M_{b2}\right)=\left[100010000\right|001]$. By steps (20–21), we add $M_{D1}=\left[100010000\right]$ to ${\mathcal{M}}_D$ and $(M_{b2},(\varnothing ,\left[001\right]),M_{D1})$ to ${\Delta }_D$. However, $M_{b3}=\left[001001001\right]$ is defined as a dead basis marking (DBM). Since $Y_T\left(M_{b3}\right)=\varnothing$, we tag $M_{b3}$ as an old node and execute step 2 to check the existence of marking without a tag in ${\mathcal{M}}_B$. Considering Theorem 1, there is no need to check the safeness of the implicit reaches of $M_{b3}$. Similarly to $M_{b2}$, at $M_{b5}$ we have $\beta \ne \varnothing$ and its value is $\beta =\left[000001102\right|010]$. Thus a new node $M_{d2}=\left[000001102\right]$ and a new relation $(M_{b5},(\varnothing ,\left[010\right]),M_{d2})$ will be added to ${\mathcal{M}}_D$ and ${\Delta }_D$, respectively.

8. Experimental Results

In this section, we will introduce the experimental results of this work. A flexible manufacturing system is used to illustrate the proposed deadlock characterization policy for labeled Petri nets. The Petri net model of the system shown in Figure 12 is a pure and bounded Petri net taken from [44]. In the net, there are 13 places $\{p_0,\cdots ,p_{12}\}$ and 10 transitions $\{t_0,\cdots ,t_9\}$. Given the system partition $\pi =(T_E,T_I)$ with $T_E=T_o=\{t_1\left(a\right),t_2\left(b\right),t_3\left(b\right),t_4\left(a\right),t_5\left(c\right),t_8\left(c\right)\}$ and $T_I=T_{uo}=\{t_0\left(\epsilon \right),t_6\left(\epsilon \right),t_7\left(\epsilon \right),t_9\left(\epsilon \right)\}$ and the initial marking $M_0=2p_0+p_5+p_6+2p_7+p_{11}+p_{12}$, Figure 13 shows that the net has 32 reachable markings, 28 of which are legal markings depicted in solid lines and 4 of which are deadlock ones depicted in dashed lines. The set of deadlock markings is $\mathcal{D}=\{M_{D_1}=2p_0+p_5+p_6+p_8+p_9,M_{D_2}=p_0+p_2+p_6+p_8+p_9,M_{D_3}=p_2+p_3+p_8+p_9,M_{D_4}=p_0+p_3+p_5+p_8+p_9\}$.

In Figure 14, we present the basis reachability graph of the net system. As the BRG shows, the number of basis markings is $\left|\mathcal{M}\right|=11$. However, none of the dead markings appears in the BRG, i.e., for all $M\in \mathcal{D}$, $M\notin \mathcal{M}$. We construct the observed graph with Algorithm 3. The number of markings is $|{\mathcal{M}}_O|=15$. We observe that $\left|\mathcal{M}\right|\le |{\mathcal{M}}_O|\le |R(N,M_0)|$ and $\mathcal{M}\subseteq {\mathcal{M}}_O\subseteq R(N,M_0)$. From the marking $M_{b0}$ we have $\beta =\left[2000011011000\right|0,2,1,0].$ Thus the dead markings $M_{d0}$ is added to the observed graph since it represents a dangerous implicit reachable marking. As the same as $M_{b1}$, the computation of the set of DIVs and DIMs results in $\beta =\left[1010001011000\right|0,2,1,0]$ and a new dead node $M_{d1}=\left[1010001011000\right]$ is added. From the marking $M_{b2}$ we also have a dangerous implicit reach given by $M_{d2}=\left[1001010011000\right]$ since $\beta =\left[1001010011000\right|0,2,1,0].$ Finally, from the marking $M_{b4}$ we have $\beta =\left[0011000011000\right|0,2,1,0]$ and a dead observed marking $M_{d3}=\left[0011000011000\right]$ is added to the observed graph.

9. Conclusion and Future Work

This paper addresses the deadlock characterization and analysis in labeled Petri nets whose transitions can be unobservable and undistinguishable. In the first part of the paper, we show that the notion of BRG cannot define and characterize correctly and accurately the deadlock problem in an LPN. We propose a first algorithm to compute the max-cardinality of the implicit vector that can fire from a given state of the system. After that, we introduce two sufficient conditions described by linear algebraic equations that will be useful to construct the observed graph. Then, we provide an algorithm to computing the observed graph. Finally, experimental results implemented using the C # tool expose the correctness and effectiveness of the proposed solution. To move forward, future research in this framework will focus on using observed graphs to ensure the deadlock-freeness and to study the liveness enforcement of systems modeled by labeled Petri nets.