Unbounded Petri Nets Research Articles

Fault-prognosability, [formula omitted]-step prognosis and [formula omitted]-step predictive diagnosis in partially observed petri nets by means of algebraic techniques

In this paper, we explore some issues relevant to fault monitoring in discrete event systems modeled by partially observed LPNs with, possibly indistinguishable observable events and acyclic unobservable subnet. Firstly, we address the (offline) fault-prognosability analysis problem. Subsequently, we tackle the two online problems of K-step fault prognosis and K-step predictive diagnosis. We propose algebraic formulations and solutions to these problems. Namely, a necessary and sufficient condition for fault-prognosability is established based on solving an integer optimization problem. The proposed approach is applicable for bounded Petri nets. As for the K-step prognosis and K-step predictive diagnosis, algebraic approaches based on state estimation on a sliding horizon are elaborated to produce relevant verdicts. The established results for K-step prognosis and K-step predictive diagnosis are applicable for both bounded and unbounded Petri nets.

Probabilistic Reachability Prediction of Unbounded Petri Nets: A Machine Learning Method

Unbounded Petri nets (UPNs) can describe and analyze discrete event systems with infinite states (DESIS). Due to the infinite state space and the combination explosion problem, the reachability analysis of UPNs is an NP-Hard problem. The existing reachability analysis methods cannot achieve an accurate result at reasonable costs (computational time and space) due to the finite reachability tree with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega$</tex-math> </inline-formula> -numbers. Based on the idea of approximating infinite space with finite states, given some limited reachable markings of a UPN, we propose a method that can quantitatively solve the UPN’s reachability problem with machine learning. Firstly, we define the probabilistic reachability of markings and transform the UPN’s reachability problem into the prediction problem of markings. The proposed method based on positive and unlabeled learning (PUL) and bagging trains a classifier to predict the probabilistic reachability of unknown markings. Finally, to predict the markings outside the positive sample set and unlabeled sample set, an iterative strategy is designed to update the classifier. Based on seven general UPNs, the results of the experiments show that the proposed method has a good performance in the accuracy and time consumption for the UPN’s reachability problem. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In discrete event systems, the reachability problem mainly studies reachable states of the system and the relationship between states, which is the basis of the system’s states, behaviors, attributes and performance analysis. For discrete event systems with infinite states, it is hard to analyze the reachable relationship between states within a finite time due to the infinite state space and the combination explosion problem. The main motivation of the paper is to propose a method that can predict the reachable relationship between the states with a probability value within a finite time. By machine learning algorithms, the method learns the feature information of the known reachable states. The reachability of unknown states in the infinite state space can be predicted approximately. The proposed approximation method can be applied to analyze the reachability properties of general discrete event systems with infinite states, such as checking whether a fault occurs in operating systems, whether a message is delivered in communication and so on.

Detectability in Discrete Event Systems Using Unbounded Petri Nets

This paper investigated the verification of detectability for discrete event systems based on a class of partially observed unbounded Petri nets. In an unbounded net system, all transitions and partial places are assumed to be unobservable. The system administrator can only observe a few observable places, i.e., the number of tokens at these places can be observed, allowing for the estimation of current and subsequent states. The concepts of quasi-observable transitions, truly unobservable transitions, and partial markings are used to construct a basis coverability graph. According to this graph, four sufficient and necessary conditions of detectability are proposed. Correspondingly, a specific example is proposed to prove that the detectability can be verified in the unbounded net system. Furthermore, based on the conclusion of detectability, the system’s ability to detect critical states was explored by using the basis coverability graph, called C-detectability. Two real-world examples are proposed to show that the detectability of discrete event systems has not only pioneered new research methods, but also demonstrated that the real conditions faced by this method are more general, and it has overcome the limitations of relying only on the ideal conditions of bounded systems for verification.

Verification of Current-State Opacity in Discrete Event Systems by Using Basis Coverability Graphs

A new approach to the verification of current-state opacity for discrete event systems is proposed in this paper, which is modeled with unbounded Petri nets. The concept of opacity verification is first extended from bounded Petri nets to unbounded Petri nets. In this model, all transitions and partial places are assumed to be unobservable, i.e., only the number of tokens in the observable places can be measured. In this work, a novel basis coverability graph is constructed by using partial markings and quasi-observable transitions. By this graph, this research finds that an unbounded net system is current-state opaque if, for an arbitrary partial marking, there always exists at least one regular marking in the result of current-state estimation with respect to the partial marking not belonging to the given secret. Finally, a sufficient and necessary condition is proposed for the verification of current-state opacity. A manufacturing system example is presented to illustrate that the concept of current-state opacity can be verified for unbounded net systems.

K-diagnosability analysis of bounded and unbounded Petri nets using linear optimization

We propose an algebraic approach to investigate K-diagnosability of partially observed labeled Petri nets which can be either bounded or unbounded. Namely, a necessary and sufficient condition for K-diagnosability is established based on the resolution of an Integer Linear Programming (ILP) problem. When the system is K-diagnosable, our approach also yields the minimal value Kmin≤K that ensures Kmin-diagnosability. The value of Kmin is calculated directly, using the same ILP formulation, i.e, without testing 1,…,(Kmin−1)-diagnosability. A second K-diagnosability approach, which is derived from the first one, is also developed on a compacted horizon providing a sufficient condition for K-diagnosability. This second technique allows for reducing the system dimensionality yielding a higher computational efficiency and allowing the characterization of the length of the sequences that lead to the fault occurrence, which is necessary to perform the K-diagnosability test of the first approach.

Abstraction Based Supervisory Control for Non-Regular -Languages∗

The problem of supervisory control, in its most basic form, is well understood for regular parameters, i.e., with regular languages for plant and upper-bound language-inclusion specification. However, when dropping the requirement of regularity, the situation becomes much more involved: even decidability becomes an issue. In this paper, we propose a simplistic but yet fairly general approach in that non-regular parameters are replaced by regular substitutes. In a language based setting we provide (a) conditions such that any solution found for the substitute parameters carries over to the original problem and (b) a refinement scheme which can be applied when the substitute parameters render the control problem at hand unsolvable. For practical purposes, we further elaborate our approach for plants represented as unbounded Petri nets and (c) identify a condition under which the alternation of trial synthesis and refinement is guaranteed to terminate with success.

Online Fault Diagnosis of Discrete Event Systems Modeled by Labeled Petri Nets Using Labeled Priority Petri Nets*

The problem of fault diagnosis of discrete event systems (DESs) concerns the capacity to detect the occurrence of fault events. In this work, we present a new fault diagnoser that uses labeled priority and λ-free diagnoser Petri nets to diagnose the fault occurrences of DESs modeled by Petri nets, which may be bounded or unbounded. By analyzing the diagnoser Petri net possible states reached after firing transitions labeled by the observable events, the diagnoser can infer about the fault occurrence. In order to allow the diagnoser to analyze the possible reachable states of the λ-free diagnoser Petri net and diagnose the fault occurrence, we propose a new approach for the state estimation of λ-free Petri nets. For a class of Petri nets, including some unbounded Petri nets, the diagnoser proposed here has a structure that does not grow indefinitely due to event observations, making the diagnoser able to perform the online diagnosis of a class of Petri nets that previous works are only able to diagnose with structures that are likely to grow indefinitely for specific sequences of event observations.

Fault diagnosis in labelled Petri nets: A Fourier–Motzkin based approach

We propose techniques for fault diagnosis in discrete-event systems modelled by labelled Petri nets, where fault events are modelled as unobservable transitions. The proposed approach combines an offline and an online algorithm. The offline algorithm constructs a diagnoser in the form of sets of inequalities that capture the legal, normal and faulty behaviour. To implement the offline algorithm, we adopt the Fourier–Motzkin method for elimination of variables from these sets of inequalities. Upon observing an event, the diagnoser is used to determine whether a fault occurred or might have occurred. The occurrence of a fault can be verified by checking the observed sequence against the sets of inequalities. This approach has the advantage that the tradeoff between the size of the diagnoser and the time for computing the diagnosis is achieved. In addition, fault diagnosis in both bounded and unbounded Petri nets can be addressed.

Analysis of Unbounded Petri Net With Lean Reachability Trees

At present, no efficient method is proposed for the liveness analysis of general unbounded Petri nets (UPNs) except some of their subclasses. Our previous work presents a non-Karp-Miller finite reachability tree, i.e., lean reachability tree (LRT) to represent their markings. It faithfully expresses and folds the reachability set of an unbounded net. It can totally avoid the efforts made by the existing modified Karp-Miller trees on the expression of potentially unbounded nodes and elimination of all fake markings. By exploiting it, this paper presents a method for comprehensively analyzing the properties of general UPNs. Particularly, we reveal the repeatability of deadlock with the unfolding of some unbounded leaves in LRT and present a sufficient and necessary condition of deadlock existence. Then, LRT and some partial trees generated from it, instead of entire reachability graphs, are utilized to analyze the liveness and reversibility of general UPNs rather than some special ones. The related theoretical results are proven. A unified algorithm based on LRT for analysis of boundedness, liveness, deadlock, and reversibility of general UPNs is developed for the first time. The results of a case study show that the presented method is effective for general UPNs.

Complex Reachability Trees and Their Application to Deadlock Detection for Unbounded Petri Nets

Deadlock detection plays an important role in the analysis of system behavior. Several kinds of reachability trees have been proposed to analyze Petri net properties including deadlock freedom. However, existing reachability trees can only solve the deadlock detection problem of bounded or some special kinds of unbounded Petri nets. To increase the applicable scope of reachability trees in the deadlock detection field, this paper presents a new type of reachability trees called complex reachability tree (CRT). Different from others, transition sequences corresponding to root-started paths of cyclic CRTs are always firable from the initial marking. The proposed trees can completely solve the deadlock detection problem of ${\omega }$ -gone node-free Petri nets, but cannot guarantee to detect all the deadlocks for the other kinds of unbounded Petri nets. Their construction method and applications are presented.

Deadlock detection-oriented unfolding of unbounded Petri nets

The state-space explosion obstructs Petri nets for large-scale system analysis. Net unfolding techniques have drawn considerable attention to solve this problem. However, existing techniques are not suitable for deadlock detection of unbounded Petri nets. To address this issue, this paper presents a deadlock detection-oriented unfolding (D2U) technique toward unbounded nets. D2U has the following advantages over traditional methods of solving deadlock detection problems: (1) Each configuration-corresponding cyclic or acyclic transition sequence in D2U is firable from the initial marking; (2) D2U can completely decide the existence of deadlocks when some conditions are met; and (3) D2U contains no spurious markings, unlike traditional unfolding techniques of unbounded Petri nets. The construction method and applications of D2U to deadlock detection of unbounded nets are presented.

Corrections to “On the Decidability and Complexity of Diagnosability for Labeled Petri Nets” [Nov 17 5931-5938

This note aims to correct a complexity result in our paper that investigates the verification of diagnosability for unbounded Petri nets. The main result of [1] is Theorem IV.1, which shows that the diagnosability verification problem can be reduced to a Petri net model checking problem called Yen's problem [2]. This main reduction is still correct. However, it was pointed out by Atig and Habermehl [3] that Yen's original complexity result is not completely correct 1. Specifically, they showed that: (i) in general, Yen's problem is still decidable, but it is as hard as the reachability problem, which corrects the original EXPSPACE-completeness claim in [2]; and (ii) for a special case called the increasing fragment, Yen's problem can be solved in EXPSPACE.

New reachability trees for analyzing unbounded Petri nets with semilinear reachability sets

New reachability trees for analyzing unbounded Petri nets with semilinear reachability sets

Lean Reachability Tree for Unbounded Petri Nets

Elaborate efforts have been made to eliminate fake markings and refine ${\omega }$ -markings in the existing modified or improved Karp–Miller trees for various classes of unbounded Petri nets since the late 1980s. The main issues fundamentally are incurred due to the generation manners of the trees that prematurely introduce some potentially unbounded markings with ${\omega }$ symbols and keep their growth into new ones. Aiming at addressing them, this work presents a non-Karp–Miller tree called a lean reachability tree (LRT). First, a sufficient and necessary condition of the unbounded places and some reachability properties are established to reveal the features of unbounded nets. Then, we present an LRT generation algorithm with a sufficiently enabling condition (SEC). When generating a tree, SEC requires that the components of a covering node are not replaced by ${\omega }$ symbols, but continue to grow until any transition on an output path of an unbounded place has been branch-enabled at least once. In return, no fake marking is produced and no legal marking is lost during the tree generation. We prove that LRT can faithfully express by folding, instead of equivalently representing, the reachability set of an unbounded net. Also, some properties of LRT are examined and a sufficient condition of deadlock existence based on it is given. The case studies show that LRT outperforms the latest modified Karp–Miller trees in terms of size, expressiveness, and applicability. It can be applied to the analysis of the emerging discrete event systems with infinite states.

An Algorithm of Recognizing Unbounded Petri Nets With Semilinear Reachability Sets and Constructing Their Reachability Trees

Reachability is one of the significant properties of Petri nets (PNs). For unbounded PNs, how to determine their reachability is an open issue. In this paper, we propose an algorithm that determines whether the reachability set of an unbounded PN is semilinear. Moreover, in the case that the unbounded PN has semilinear reachability set, the proposed algorithm can construct a new reachability tree (NRT) that exactly characterizes its reachability set. In addition, it can be decided based on NRT whether an unbounded PN suffers from deadlocks. The results are illustrated via examples.

On the Decidability and Complexity of Diagnosability for Labeled Petri Nets

In this paper, we investigate the decidability and complexity of the fault diagnosis problem in unbounded labeled Petri nets. First, we show that checking diagnosability for unbounded Petri nets is decidable. We present a new necessary and sufficient condition for diagnosability, which can be reduced to a model checking problem for unbounded Petri nets. Then, we show that checking diagnosability for unbounded Petri nets is EXPSPACE-complete. This complexity result is further extended to various subclasses of Petri nets. To the best of our knowledge, this is the first paper that establishes decidability and complexity results for diagnosability of unbounded Petri nets.

Liveness Analysis of ω-Independent Petri Nets Based on New Modified Reachability Trees

Liveness of Petri nets means all activities in a modeled system can potentially take place and thus implies deadlock freedom. The research on liveness analysis approaches is inadequate. This paper proposes a liveness judgment reachability graph (LJRG) approach to analyze the liveness of $ {\omega }$ -independent Petri nets. Such nets can be loosely explained as a class of unbounded Petri nets in which the changes of tokens in unbounded places are not related to each other. This paper proposes several algorithms to transform a new modified reachability tree to a new modified reachability graph and then transform it to an LJRG. It then develops the application of LJRG into the liveness analysis of $ {\omega }$ -independent unbounded Petri nets. The proposed method provides a new theoretical method and important tool for the liveness analysis of unbounded Petri nets. It is illustrated via some examples.

A reduced reachability tree for a class of unbounded petri nets

As a powerful analysis tool of Petri nets, reachability trees are fundamental for systematically investigating many characteristics such as boundedness, liveness and reversibility. This work proposes a method to generate a reachability tree, called ωRT for short, for a class of unbounded generalized nets called ω-independent nets based on new modified reachability trees (NMRTs). ωRT can effectively decrease the number of nodes by removing duplicate and ω-duplicate nodes in the tree, and verify properties such as reachability, liveness and deadlocks. Two examples are provided to show its superiority over NMRTs in terms of tree size.

A Survey of Reachability Trees of Unbounded Petri Nets

In recent years both industry and academia have paid much attention to the theory and applications of Petri nets. Reachability is a basic property of a Petri net, and many properties can be analyzed via it. However, analyzing the reachability problem of unbounded Petri nets by finite reachability trees has been an open problem since the inception of Petri nets. Researchers began to study the problem of reachability trees over 40 years ago. However, they made only limited progress over the last 20 years due to its complexity and difficulty. We present an overview of some important contributions toward its solution. The focuses are on four novel finite reachability trees: finite reachability tree(FRT),augmented reachability tree(ART), modified reachability tree(MRT) and new modified reachailbity tree(NMRT). The paper concludes with a discussion of directions for future research of the reachability problem of unbounded Petri nets.

Macro liveness graph and liveness of ω-independent unbounded nets

Liveness is a basic property of a system and the liveness issue of unbounded Petri nets remains one of the most difficult problems in this field. This work proposes a novel method to decide the liveness of a class of unbounded generalized Petri nets called ω-independent unbounded nets, breaking the existing limits to one-place-unbounded nets. An algorithm to construct a macro liveness graph (MLG) is developed and a critical condition based on MLG deciding the liveness of ω-independent unbounded nets is proposed. Examples are provided to demonstrate its effectiveness.