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.

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

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 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

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

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

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.

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.

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.

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.