Coverability Graph Research Articles

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.

Commodification of accelerations for the Karp and Miller Construction

Karp and Miller’s algorithm is based on an exploration of the reachability tree of a Petri net where, the sequences of transitions with positive incidence are accelerated. The tree nodes of Karp and Miller are labeled with ω-markings representing (potentially infinite) coverability sets. This set of ω-markings allows us to decide several properties of the Petri net, such as whether a marking is coverable or whether the reachability set is finite. The edges of the Karp and Miller tree are labeled by transitions but the associated semantic is unclear which yields to a complex proof of the algorithm correctness. In this work we introduce three concepts: abstraction, acceleration and exploration sequence. In particular, we generalize the definition of transitions to ω-transitions in order to represent accelerations by such transitions. The notion of abstraction makes it possible to greatly simplify the proof of the correctness. On the other hand, for an additional cost in memory, which we theoretically evaluated, we propose an “accelerated” variant of the Karp and Miller algorithm with an expected gain in execution time. Based on a similar idea we have accelerated (and made complete) the minimal coverability graph construction, implemented it in a tool and performed numerous promising benchmarks issued from realistic case studies and from a random generator of Petri nets.

Stop-transitions of Petri Nets*

A transition t stops a place/transition Petri net if each reachable marking of the net enables only finite occurrence sequences without occurrences of t (i.e., every infinite occurrence sequence enabled at this marking contains occurrences of t). Roughly speaking, when t is stopped then all transitions of the net stop eventually. This contribution shows how to identify stoptransitions of unbounded nets using the coverability graph. Furthermore, the developed technique is adapted to a more general question considering a set of stop-transitions and focussing on a certain part of the net to be stopped. Finally, an implementation of the developed algorithm is presented.

Correction to “Synchronizing sequences on a class of unbounded systems using synchronized Petri nets”

The paper mentioned in the title proposed a new definition of increasing repetitive input sequences for synchronized Petri nets and stated that it can be used to construct a finite modified coverability graph for any unbounded SynPN. In this note, we discuss the notion of unboundedness for SynPNs and show via an example that actually the modified coverability graph may be infinite due the presence of increasing sequences that are not repetitive.

K-Codiagnosability Verification of Labeled Petri Nets

Fault detection is quite important for discrete event systems. We investigate K-codiagnosability of Petri nets in this paper under the framework that some local sites monitor the operation of the system using their own masks. They exchange information with a coordinator while do not communicate with each other. A fault is detected when there exists a site can diagnose it. We recall the notion of Modified Verifier Nets (MVNs), and prove that K-codiagnosability can be verified looking at some special cycles in the reachability or coverability graph of the MVN. In particular, the proposed approach is available for bounded and unbounded nets. Finally, we give an algorithm to compute the minimum value of K.

On–The–Fly Diagnosability Analysis of Bounded and Unbounded Labeled Petri Nets Using Verifier Nets

Abstract This paper considers the problem of diagnosability analysis of discrete event systems modeled by labeled Petri nets (LPNs). We assume that the LPN can be bounded or unbounded with no deadlock after firing any fault transition. Our approach is novel and presents the on-the-fly diagnosability analysis using verifier nets. For a given LPN model, the verifier net and its reachability graph (for a bounded LPN) or coverability graph (for an unbounded LPN) are built on-the-fly and in parallel for diagnosability analysis. As soon as a diagnosability decision is established, the construction is stopped. This approach achieves a compromise between computation limitations due to efficiency and combinatorial explosion and it is useful to implement an engineering approach to the diagnosability analysis of complex systems.

Conversion of 1-Place-Unbounded Synchronized Petri Nets into Weighted Automata

Testing is a fundamental technique for system design and verification to ensure security and reliability. However its application to discrete event systems modeled by unbounded synchronized Petri nets is not straightforward because there exists no finite exact representation for the infinite state space of these models. In this paper, we consider a special class of synchronized Petri nets, called 1-place-unbounded, that contain a single unbounded place. We show how a coverability graph that precisely describes the state space of such a model can be constructed extending to synchronized nets a technique previously presented for place/transition nets. In addition, this algorithm can also be used to verify whether a given synchronized Petri net contains exactly a single unbounded place. Next, we show that any net belonging to this special class can be converted into an equivalent weighted automaton. Based on this conversion, we observe that the testing of 1-place-unbounded synchronized Petri nets can be approached using the methods and results existing in the literature for the testing of weighted automata.

Tools and Methods for RTCP-Nets Modeling and Verification

Abstract RTCP-nets are high level Petri nets similar to timed colored Petri nets, but with different time model and some structural restrictions. The paper deals with practical aspects of using RTCP-nets for modeling and verification of real-time systems. It contains a survey of software tools developed to support RTCP-nets. Verification of RTCP-nets is based on coverability graphs which represent the set of reachable states in the form of directed graph. Two approaches to verification of RTCP-nets are considered in the paper. The former one is oriented towards states and is based on translation of a coverability graph into nuXmv (NuSMV) finite state model. The later approach is oriented towards transitions and uses the CADP toolkit to check whether requirements given as μ-calculus formulae hold for a given coverability graph. All presented concepts are discussed using illustrative examples

A Smart Home Context-aware Model Based On UML and Colored Petri Net

Smart Home is one of the main application domains of Pervasive Computing, and it can provide Context-aware Services to homeowners. As context information is obtained from different sources in the Smart Home environment, the context information may be heterogeneous, inaccurate and dynamic. How to effectively model these different types of context information is a main research issue. To address this issue, we propose a hybrid Context-aware Modeling approach based on UML and Colored Petri Net in this paper. We also model leave home scenario using this approach, and use Coverability Graph to verify the leave home scenario model.

Diagnosability of Petri nets with observation graphs

This paper addresses the problem of diagnosability for dynamic discrete event systems modeled with bounded or unbounded Petri nets that are deadlock-free and monitored with sensor configurations with marking and event measurements. The proposed method gives necessary and sufficient conditions for diagnosability. It is based on the transformation of the coverability graph into an observation graph that encodes all observation sequences of measured markings and events with respect to the sensor configuration. This graph also encodes all sequences of transitions that may fire from any reachable marking of the Petri net. Diagnosability is determined by analyzing the paths and circuits in the observation graph. The method is illustrated with several examples of bounded or unbounded Petri nets.

Coverability Graph of Fuzzy Interpreted Petri Net

This paper describes a coverability graph that can be used to analyze properties of fuzzy interpreted Petri net. It also proposes a definition of the graph and a detailed algorithm for its construction. An example graph of the net is shown. The results contribute to expanding the existing options of net analysis at the design stage.

Identification of one-place-unbounded Petri nets from their modified coverability graph

This paper presents the problem of identifying a one-place unbounded Petri net. Given an unlabelled graph that represents the modified coverability graph of a net, we establish a Petri net model whose modified coverability graph is isomorphic to the unlabelled graph, and that can identify the weight of the arcs that cannot be obtained from the coverability graph of the net. Based on the partition of the nodes in the unlabelled graph, we guess and decide the structure of a Petri net and the weight of the arcs by an integer linear programming problem. The unknowns to be determined are the elements of the pre- and post-incidence matrices and the initial marking of the net, which can be computed by solving an integer linear programming problem. Finally, an example is used to validate the rationality and effectiveness of the proposed approach.

A Petri-net and QoS Based Model for Automatic Web Service Composition

Web services are widely used because of their features of interoperability, loose-coupled and platform independent. Web services composition is one of the most popular topics in service computing area. In this paper a method based on Petri net coverability problem and utility of Web services is proposed to handle automatic service composition. The former is for satisfying functional requirements of service composition while the latter corresponds to non-functional properties. For a set of given services, each service is translated into component of Petri net and the input/output places with same semantics are merged. According to user’s input and desired output, the initial marking and target marking can be obtained. Then the coverability tree and coverability graph can be constructed. Next, the nodes on the coverability graph which can cover the target marking should be find out. If there are more than one coverability paths, we determine the smallest weight denoted as “SW” on each path. The weight is defined as the utility calculated according to the QoS. At last, services on the path with largest SW are selected as service executing sequence to reach the goal of automatic service composition.

A formal method for analysing and assessing operational risk in supply chains

This paper formalises an objective method to identify and assess operational risk in supply chains. The proposed approach exploits the analogy among logistics networks and dynamical systems. In particular, risky events identification is based on the analysis of the coverability graph of the timed attributed Petri net describing the analysed supply chain, whereas risk assessment is done by building the simulation model of the studied logistic network, experimenting on it and applying ANOVA to analyse the results and evaluating the importance among the risky events previously figured out. The method has been applied to an example case represented by a single-item, three-stage supply chain.

Checking system boundedness using ordinary differential equations

Boundedness is one of the most important properties of discrete Petri nets. Determining the boundedness of a Petri net is usually done through building coverability graph or coverability tree. However, the computation is infeasible for complex applications because the size of the coverability graph may increase faster than any primitive recursive functions. This paper proposes a new technique to check the boundedness without causing this problem. Let a concurrent system be represented by a (discrete) Petri net. By relaxing the (discrete) Petri net to a continuous Petri net, we can model the concurrent system by a family of ordinary differential equations. It has been shown that the boundedness of the discrete Petri net is equivalent to the boundedness of the solutions of the corresponding ordinary differential equations. Hence, we can check the boundedness of a (discrete) Petri net by analyzing the solutions of a family of ordinary differential equations. A case study demonstrates the benefits of our technique.

Incremental construction of coverability graphs

Incremental construction of coverability graphs

Modeling and Analyzing of Workflow Authorization Management

The administration of authorizations in an organization is a complex task. To ensure that tasks constituting the business processes are performed by authorized users, a proper authorization mechanism is required. Alturi and Huang have proposed a workflow authorization model and presented a color-timed Petri net based representation of their model. In this paper, we extend their model by using the colored Petri net formalism to model authorization management, security constraints like separation of duties, and role hierarchy in an elegant way to establish an integrated authorization management model. One of the great advantages of using Petri net formalism for system modeling is its strong mathematical foundation and the availability of a rich set of analysis techniques. Therefore, we will show in this paper the use of linear algebraic technique to analyze the reachable authorization states, and coverability graph to calculate the valid execution chains against the colored Petri net based workflow authorization management model.

Observability of place/transition nets

We discuss the problem of estimating the marking of a place/transition (P/T) net based on event observation. We assume that the net structure is known while the initial marking is totally or partially unknown. We give algorithms to compute a marking estimate that is a lower bound of the actual marking. The special structure of Petri nets allows us to use a simple linear algebraic formalism for estimate and error computation. The error between actual marking and estimate is a monotonically nonincreasing function of the observed word length, and words that lead to null error are said to be complete. We define several observability properties related to the existence of complete words, and show how they can be proved. To prove some of them, we also introduce a useful tool, the observer coverability graph, i.e., the usual coverability graph of a P/T net augmented with a vector that keeps track of the estimation error on each place of the net. Finally, we show how the estimate generated by the observer may be used to design a state feedback controller for forbidden marking specifications.

Decidability problems in grammar systems

Most of the basic decision problems concerning derivations in cooperating distributed grammar systems have so far been open, possibly because of the lack of unifying methods and techniques. In this paper such a unifying device is proposed. It is called a coverability tree because it bears some resemblance to the coverability graph of place/transition Petri nets and vector addition systems. The coverability tree is always finite, which leads to rather strong decidability properties concerning both arbitrary and terminal derivations. Our method is largely independent of the mode of the derivations and answers most of the direct decidability questions about the components of the system.