Showing papers on "Petri net published in 2023"

A Learning-Embedded Attributed Petri Net to Optimize Student Learning in a Serious Game

Abstract: Serious games (SGs) are a practice of growing importance due to their high potential as an educational tool for augmented learning. However, little effort has been devoted to address student learning optimization in an SG from a systematic point of view. This article tackles this challenge by developing a learning-embedded attribute Petri net (LAPN) model to represent game flow and student learning decision-makings. The dynamics of learner behaviors in game are then addressed through the incorporation of learning mechanisms (i.e., reinforcement learning (RL) and random forest classification) into the Petri net model for knowledge reasoning and learning. Finally, an algorithm based on LAPN is proposed, aiming to guide learners to achieve a faster and better solution to problem-solving in game. The benefit of the proposed model and algorithm is then demonstrated in the SG Gridlock.

Hierarchical petri net modeling for system dynamics and control of manufacturing systems

Abstract: The objective of this paper is to introduce a Petri net-based modeling structure with a state space representation for manufacturing systems within the context of hierarchical control. A Flexible Manufacturing Cell (FMC) having parallel processing capability and a material handling system with limited capability is investigated. A machine-oriented timed colored Petri net (TCPN) modeling technique is introduced to represent both sequential and parallel processes characteristics of the cell. Based on a TCPN model at the cell level, a workstation level model is configured through a top-down decomposition approach resulting in a hierarchical multilevel PN model. The dynamical system representation of the TCPN model at the cell level is transformed into a mathematically based state space model. The control for the FMC is formulated within a discrete-time optimal control framework with varying time intervals and a time-variant state equation. A scheduling heuristic solution and associated performance evaluation results are discussed. The state space representation at the workstation level is also discussed to demonstrate the continuity of the proposed modeling structure.

Cross-Department Collaborative Healthcare Process Model Discovery From Event Logs

Abstract: Healthcare plays an increasingly essential role in our daily life. Modern Hospital Information Systems (HISs) record and store detailed medical treatment process information for all patients as event logs. By taking event logs as input, process mining techniques have been widely applied to extract valuable insights to improve medical treatment processes and deliver better healthcare services. However, considering the complexity of collaborations among different medical departments, existing model discovery techniques cannot be applied directly. To handle this limitation, this paper proposes a novel approach to support the discovery of Cross-department Collaborative Healthcare Process (CCHP) models from medical event logs. Specifically, an extension of classical Petri Nets with message and resource attributes is first introduced to formalize CCHPs. Then, a novel discovery algorithm is proposed to discover Intra-department Healthcare Process (IHP) models. Next, collaboration patterns among medical departments are formalized and corresponding discovery algorithms are given on that basis. Finally, a global CCHP model is obtained by integrating all discovered collaboration patterns and IHP models. By using four public medical event logs, we quantitatively compare our approach with the state-of-the-art process mining techniques in terms of model quality, and our experimental results demonstrate that the proposed approach can discover more accurate healthcare process models. Note to Practitioners —The recorded medical event logs by HISs can be used to extract valuable insights for the analysis of healthcare processes. However, existing process model discovery techniques cannot be applied for the analysis directly due to the complex collaborations among different medical departments of a hospital. This paper introduces a novel approach for cross-department collaborative healthcare process model discovery from medical event logs. All proposed techniques are fully implemented and publicly available. Using four public medical event logs, we show the applicability and advantages of our approach against existing ones. The proposed techniques are applicable to the model discovery and behavior understanding of real-life operational healthcare processes.

Formal Modeling and Discovery of Hierarchical Business Processes: A Petri Net-Based Approach

Abstract: Business processes are critical for information systems to control workflows and deliver services. Although existing process discovery techniques can generate flat process models from business event logs, few of them have investigated the notion of hierarchy (i.e., subprocesses) yet. To fill the gap, this article first defines the concept of hierarchical Petri nets (HPNs), which can support the formal modeling and correctness verification of processes with subprocesses. Followed by that, we propose an approach which can effectively discover HPNs from event logs with lifecycle information. Moreover, to quantify the quality of discovered HPNs, details on how to transform an HPN to a classical Petri net are given such that existing metrics can be applied. All proposed approaches have been fully implemented in ProM, and experiments over both synthetic and real-life event logs demonstrate that our approach can effectively discover hierarchical process models. Specifically, compared to exiting approaches on processes discovery, our approach can generally perform better in terms of model quality.

Critical Observability of Labeled Time Petri Net Systems

Abstract: A time Petri net is said to be critically observable at a given time instant if the markings consistent with any observation at the time instant are included either in the set of critical markings or non-critical markings. This work studies the verification problem of critical observability for timed discrete event systems modeled by bounded labeled time Petri nets. The proposed method is a two-fold process: a preliminary verification of critical observability for the underlying logic labeled Petri net and a further verification considering the time constraint associated with each transition. The first step is based on the concurrent composition of a reachability graph of the logic net. If the logic net is critically observable, then the time net is also critically observable at any given time instant. Otherwise, the second step is to design an algorithm to compute all pairs of transition-class sequences that violate critical observability at the given time instant, and then a set of linear programming problems is exploited to check critical observability for the corresponding timed system. Note to Practitioners —Timed discrete event systems provide a theoretical model for safety-critical real applications such as air traffic management, smart grid, and industrial control, which are vulnerable to malicious attack and destruction at some particular time instants such that a system may be misled to dangerous states. Critical observability of a timed discrete event system is a property with which the predefined dangerous states can be determined and detected from the observation by an observer at a given time instant. This research aims to offer a systematic approach to check critical observability for timed discrete event systems modeled by bounded labeled time Petri nets, which can also present some new ideas and insights for practitioners in the field of safety-critical systems.

Optimal Multi-Robot Path Planning for Cyclic Tasks using Petri Nets

Abstract: In this paper, we investigate the problem of optimal multi-robot path planning for a cyclic task represented by a particular type of linear-temporal logic (LTL) formulae. Specifically, the team of robot needs to fulfill a given LTL formula and at the same time, accomplishes some particular tasks infinitely often. To avoid the state-space explosion when the number of robot increases, we use Petri nets to model the team of multi-robot. Our goal is to find an optimal infinite sequence in the prefix-suffix form for each robot such that the average cost per task is minimized. We propose an efficient planning method based on the notion of basis reachability graph (BRG), which is a compact representation of the reachability space of the PN. We demonstrate the computational efficiency of our method through illustrative examples.

Fault Detection, Diagnostics, and Treatment in Automated Manufacturing Systems Using Internet of Things and Colored Petri Nets

Abstract: Internet of things (IoT) applications, which include environmental sensors and control of automated manufacturing systems (AMS), are growing at a rapid rate. In terms of hardware and software designs, communication protocols, and/or manufacturers, IoT devices can be extremely heterogeneous. Therefore, when these devices are interconnected to create a complicated system, it can be very difficult to detect and fix any failures. This paper proposes a new reliability design methodology using “colored resource-oriented Petri nets” (CROPNs) and IoT to identify significant reliability metrics in AMS, which can assist in accurate diagnosis, prognosis, and resulting automated repair to enhance the adaptability of IoT devices within complicated cyber-physical systems (CPSs). First, a CROPN is constructed to state “sufficient and necessary conditions” for the liveness of the CROPN under resource failures and deadlocks. Then, a “fault diagnosis and treatment” technique is presented, which combines the resulting network with IoT to guarantee the reliability of the CROPN. In addition, a GPenSIM tool is used to verify, validate, and analyze the reliability of the IoT-based CROPN. Comparing the results to those found in the literature shows that they are structurally simpler and more effective in solving the deadlock issue and modeling AMS reliability.

Petri Nets-Based Modeling Solution for Cyber–Physical Product Control Considering Scheduling, Deployment, and Data-Driven Monitoring

Abstract: For a complex electromechanical product that is a cyber–physical system (CPS), its dynamic behaviors are embodied in the closed-loop control between the logic process in its cyber component and actual actuators/sensors in its physical component, and thus, a well-defined model of the control is important to create a digital twin that acts as much like the real machine as possible. This article proposes a Petri nets (PNs)-based modeling solution that employs hybrid PNs (HPNs) for physics and system of sequential systems with shared resources (S4R) nets for logic in building a hierarchical control model. We also present PNs technologies for implementing a smooth transition and bidirectional mapping from the virtual prototype to the real machine. These technologies involve a PNs integration of a reinforcement learning (RL) method for generating a workflow scheduling agent in design, an extension of PNs definitions that is compatible with the microcontroller for easy deployment in manufacturing, and an architecture of PNs execution recording for data-driven monitoring in service. A software kit is provided for the solution that includes an integrated development environment of PNs, tools for quickly building a virtual prototype, and a monitor server for remote data-driven monitoring. This solution is successfully applied in the development of a typical cyber–physical product case, namely, the chemiluminescence immunoassay (CLIA) analyzer.

On Quantitative Properties Preservation in Reconfigurable Generalized Stochastic Petri Nets

Abstract: Generalized stochastic Petri nets (GSPNs) have been extended to several dynamic-structure formalisms providing suitable tools for the modeling and verification of reconfigurable discrete-event systems (R-DESs). However, analyzing the performance of large-complex R-DESs remains a big challenging issue. Indeed, dynamic-structure GSPNs still rely on old-fashioned techniques often causing the state-space explosion problem. In this article, we present a new technique for the quantitative analysis of a dynamic-structure formalism called reconfigurable GSPNs without computing the whole state space. This work describes new reconfiguration forms used to preserve desired quantitative properties of parts of interest after each reconfiguration. Therefore, it is only required to verify the examined properties at an initial configuration. The proposed technique is proven to effectively reduce the state space and shorten the computation time in such cases. Finally, some experimental results are provided to illustrate that, from a computational perspective, the developed approach outperforms the existing tools.

A formal GSPN model of a virtual doctor dialogue system

Abstract: This paper presents a formal generalized stochastic Petri net (GSPN)–based modeling of a virtual doctor (VDr) dialogue system. The interaction model between the VDr dialogue system and the human patient is a modified version of the Wickens model. The purpose of this model is to provide a better understanding of the interaction between the VDr dialogue system and the patient, as well as the integration of other components in the system, such as the patient’s medical history and emotions. Initially, the formal definition of the GSPN model of the VDr dialogue system is presented by incorporating the token color concept, which is a feature of Color GSPNs. Then, the GSPN formal modeling is described. Note that, the GSPN model presented here is at the simulation level and not a product. Furthermore, some examples are provided with the intention of facilitating the understanding of the token color utilization and the different GSPN markings.

Detection of Actuator Enablement Attacks by Petri Nets in Supervisory Control Systems

Abstract: The feedback control system with network-connected components is vulnerable to cyberattacks. We study a problem of attack detection in supervisory control of discrete-event systems. The scenario of a system subjected to actuator enablement attacks is considered in this article. We also consider that some unsafe places that should be protected from an attacker exist in the system, and some controllable events that are disabled by a supervisor might be re-enabled by an attacker. This article proposes a defense strategy to detect actuator enablement attacks and disable all controllable events after detecting an attack. We design algorithmic procedures to determine whether the system can be protected against damage caused by actuator enablement attacks, where the damage is predefined as a set of “unsafe” places. In this way, the system property is called “AE-safe controllability”. The safe controllability can be verified by using a basis diagnoser or a basis verifier. Finally, we explain the approach with a cargo system example.

A stochastic Petri Net‐based approach for operational performance estimation of quay cranes

Abstract: Reliability, availability, and maintainability (RAM) of quay cranes (QCs) are essential for an effective port operation. This study estimates operational RAM of QCs using the Stochastic Petri Net (SPN) modelling. Asset Performance Assessment (APA) was performed to conduct the study by a SPN model. Based on the operation and maintenance data, probability distributions of Mean Time Between Failure (MTBF) and Mean Time to Repair (MTTR) is determined by Goodness of Fit Test of Anderson Darling. Subsequently, these distributions are applied in APA to determine the frequencies of failure/breakdowns, duration of downtimes, availability, and reliability with the assistance of the SPN model and Monte Carlo Simulation. The availability of the analysed QC is 0.97 or 97%. The results of this work are verified by the comparison with historical data. The outcomes of this paper contribute to reliability, availability, maintainability, and risk management of QCs.

Implicit Reward Structures for Implicit Reliability Models

Abstract: A new methodology for effective definition and efficient evaluation of dependability-related properties is proposed. The analysis targets the systems composed of a large number of components, each one modeled implicitly through high-level formalisms, such as stochastic Petri nets. Since the component models are implicit, the reward structure that characterizes the dependability properties has to be implicit as well. Therefore, we present a new formalism to specify those reward structures. The focus here is on component models that can be mapped to stochastic automata with one or several absorbing states so that the system model can be mapped to a stochastic automata network with one or several absorbing states. Correspondingly, the new reward structure defined on each component's model is mapped to a reward vector so that the dependability-related properties of the system are expressed through a newly introduced measure defined starting from those reward vectors. A simple, yet representative, case study is adopted to show the feasibility of the method.

SMACS: A framework for formal verification of complex adaptive systems

Abstract: Abstract Self-adaptive systems (SASs) have the capability to evaluate and change their behavior according to changes occurring in the environment. Research in this field is being held since mid-60, and over the last decade, the importance of self-adaptivity is being increased. In the proposed research, colored petri nets (CPN) formal language is being used to model self-adaptive multiagent system. CPN is increasingly used to model self-adaptive complex concurrent systems due to its flexible formal specification and formal verification behavior. CPN being visually more expressive than simple, Petri Nets enable diverse modeling approaches and provides a richer framework for such a complex formalism. The main goal of this research is to apply self-adaptive multi-agent concurrent system (SMACS) for complex architectures. In our previous research, the SMACS framework is proposed and verified through traffic monitoring system. All agents of SMACS are also known as intelligent agents due to their self-adaptation behavior. Due to decentralized approach in this framework, each agent will intelligently adapt its behavior in the environment and send updates to other agents. In this research, we are choosing smart computer lab (SCL) as a case study. For internal structure of each agent modal, μ \mu -calculus will be used, and then a model checker TAPAs: a tool for the analysis of process algebras will be applied to verify these properties. CPN-based state space analysis will also be done to verify the behavioral properties of the model. The general objective of the proposed system is to maximize the utility generated over some predetermined time horizon.

Small scale IoT device privacy evaluation using Petri net modeling

Abstract: Most Small Scale IoT (SSIoT) devices on the market gather a significant amount of sensitive information, yet many lack privacy controls, introducing significant privacy and safety risk to users. Such risks stem from the lack of privacy integration into the system development process. No formalized SSIoT data flow model currently integrates privacy elements for evaluation during the system development lifecycle (SDLC). This work aims to review current data flow modeling techniques, used in most SSIoT System Development Lifecycle (SDLC), to identify privacy gaps and assess requisite privacy controls necessary to improve user privacy. To verify this, we designed a simulation experiment using Petri net to evaluate the current privacy controls and hotspots during SSIoT data transitions. We assess our Petri net model using a Barbie Smart connected toy user interaction. The results show that Petri net has unique privacy elements and verification schemes over all other data flow modeling techniques. Further, it provides privacy assurance, evaluates privacy by identifying privacy hotspots needing controls, and minimizes privacy-related risks such as breach of personally identifiable information and interaction data during SSIoT device use.

Checking Missing-Data Errors in Cyber-Physical Systems Based on the Merged Process of Petri Nets

Abstract: Missing-data errors easily occur in cyber-physical systems (CPSs). Although many business process modeling notation (BPMN)-based methods are proposed to model CPSs and detect errors, it is hard to automatically verify their correctness, especially in the data flows, due to their lack of formal specifications. By comparison, Petri nets, as a formal method, are widely used to detect data-flow errors. However, these methods easily suffer from the state-space explosion problem. This is mainly because their reachability graphs or state transition graphs are based on the interleaving semantics. As an unfolding technique of Petri net, a merged process can characterize concurrency relations and alleviate this problem. Thus, we utilize the merged process of Petri net with data (PD-net) to check the missing-data errors of the CPS. We first transform a BPMN of the CPS into a PD-net and generate its merged process. Meanwhile, we analyze its structural behaviors and data-adjacent events. Furthermore, we propose an algorithm for checking missing-data errors. In addition, a case study and some experiments are done to show the practicality and effectiveness of our method.

Validation of Industrial Automation Systems Using a Timed Model of System Requirements

Abstract: Validation of industrial automation systems is the process of checking that commissioner requirements are successfully implemented. Formal approaches are needed when the considered system is critical. The method presented in this article relies on a model-based approach that exploits the compactness and graphical representation of time-interpreted Petri nets, which adds input–output interpretation to transitions/places and embeds time information. These nets are here used with multiple-server semantic to allow effective modeling of typical automation system requirements. The key idea of the system validation approach is to compare the observed behavior of the automation system with the expected behavior, as generated by updating online the model of system requirements using a state estimation algorithm. Also, an off-line procedure is provided to evaluate the evolutions admitted by the model but not observed. Both procedures yield useful data to the validation engineer, allowing to speed up the validation process. Technological issues due to the synchronous nature of controllers and the implications of their programming are considered.

Performance Optimization for a Class of Petri Nets

Abstract: Petri nets (PNs) are widely used to model flexible manufacturing systems (FMSs). This paper deals with the performance optimization of FMSs modeled by Petri nets that aim to maximize the system’s performance under a given budget by optimizing both quantities and types of resources, such as sensors and devices. Such an optimization problem is challenging since it is nonlinear; hence, a globally optimal solution is hard to achieve. Here, we developed a genetic algorithm combined with mixed-integer linear programming (MILP) to solve the problem. In this approach, a set of candidate resource allocation strategies, i.e., the choices of the number of resources, are first generated by using MILP. Then, the choices of the type and the cycle time of the resources are evaluated by MILP; the promising ones are used to spawn the next generation of candidate strategies. The effectiveness and efficiency of the developed methodology are illustrated by simulation studies.

Reliability and Performance Measurement of Safety-Critical Systems Based on Petri Nets: A Case Study of Nuclear Power Plant

Abstract: Safety-critical systems (SCSs) mitigate the risk of catastrophic loss of assets and hence do have high dependability targets. Performance and reliability are the critical dependability attributes, particularly in control and safety systems, and hence essential to measure to ensure the dependability. Traditional methods either are not capable to capture the system dynamics or encounter state explosion problem. Also, the methods are not able to measure all critical performance attributes. This article proposes a novel approach to measure the performance and reliability of SCSs. Such systems contain multiple interconnecting processing nodes, the functional requirements of which are modeled using Petri net (PN). A set of ordinary differential equations (ODEs) is derived from the PN model that represents the state of the system. The ODE solution can be used to measure the critical performance attributes, such as latency time and throughput of the system. The proposed method can avoid the state explosion problem and also introduces new metrics of performance, along with their measurement: deadlock, liveness, stability, boundedness, and steady state. The proposed technique is applied to a case study of nuclear power plant. We obtained 99.887% and 99.939% accuracy of performance and reliability measurement, respectively, which proves the effectiveness of our approach.

MODI: A Structured Development Process of Mode-Based Control Algorithms in the Early Design Stage of Building Energy Systems

Abstract: The growing share of renewable energy sources in building energy systems leads to more complex energy conversion and distribution systems. The current process of developing appropriate control functions for energy systems is insufficient and consequently error-prone. Regarding this problem, a new method is expected to systematically develop appropriate control functions for buildings and reduce design errors in this process. This paper introduces the MODI method, aiming at a structured development process of mode-based control algorithms to reduce errors in the early design stages of buildings. A complete framework and a standardized application process of the MODI method will be established to systematically design mode-based control algorithms described through signal-interpreted Petri nets. Furthermore, we performed a simulation-assisted evaluation approach to test and improve the performance of the control algorithms generated by MODI. In a case study, we applied MODI to develop a mode-based control strategy for an energy system containing heating and cooling supply networks. The desired control strategy was tested and tuned in a simulation phase. Compared to a reference control, the mode-based control algorithm shows an improvement in system efficiency by 4% in winter and 8% during the transitional season phase.

Dynamic Assignment of Roles and Users for Business Processes under Security Requirements

Abstract: It is very important to obtain assignment plans of roles and users for business processes or workflow processes under security requirements in enterprises. The assignment plans can help the enterprises improve the efficiency of execution as well as reduce their costs. However, the existing methods ignore the dynamics of users and roles, and cannot deal with security requirements. In this paper, we propose a new approach to solve the problem of dynamic assignment of roles and users for business processes under security requirements. First, a role and user assignment graph (RUAG) based on Petri nets is constructed to record detailed information about assignment plans. Second, the optimal composition of assignment plans meeting security requirements is extracted from RUAG for multiple concurrent business processes. Third, we analyze the changes of assignment plans during execution of business processes and present the procedure to adjust them. Compared with the existing methods, our approach can improve the accuracy of assignment plans, enhance the efficiency of dynamic assignment, and reduce the costs of enterprises.

Real-Time Adaptive Allocation of Emergency Department Resources and Performance Simulation Based on Stochastic Timed Petri Nets

Abstract: Overcrowding in emergency departments (EDs) is a common problem encountered by healthcare systems worldwide. Its essence is the imbalance between the need for emergency care and available resources, such as doctors, nurses, medical supplies, and treatment facilities and spaces. Such an imbalance increases with the volume of visiting patients. To solve the problem of ED overcrowding, service providers need to ensure rational allocation of resources in the emergency process to the greatest extent. This article uses stochastic timed Petri nets (STPNs) as a modeling and simulation tool to optimize the resource allocation in the emergency care workflow. On the basis of STPN simulation architecture, we propose a novel “observation-response” block (ORB) to adaptively supplement the corresponding resources according to the local crowding situation in the emergence workflow, so as to reduce the waiting time of patients in urgent need of treatment. In this article, models of patient arrival, triage, and examination process are constructed. Then, considering the waiting time of patients as the optimization objective, the statistical simulation based on STPN models is performed to verify the effectiveness of the proposed ORB block in the emergency workflow resource optimization process. The presented work provides a feasible way for the optimal ED resource allocation.

Reproduction of the control plane as a method of selection of settings for an adaptive fuzzy controller with Petri layer

Abstract: : The purpose of this paper is to show possibility and advantages of initial control plane reproduction for an adaptive fuzzy controller. Usually the fuzzy control is used when the object is not very well known. Yet the truth is, however, that some, at least general information about the object, is available. Usually, in such a case, optimization algorithms are used to tune the control structure. The purpose of this article is to show how to find a starting point that is closer to optimum than a statistically random point, and this way to obtain better results in a shorter time.

Stochastic Decision Petri Nets

Abstract: We introduce stochastic decision Petri nets (SDPNs), which are a form of stochastic Petri nets equipped with rewards and a control mechanism via the deactivation of controllable transitions. Such nets can be translated into Markov decision processes (MDPs), potentially leading to a combinatorial explosion in the number of states due to concurrency. Hence we restrict ourselves to instances where nets are either safe, free-choice and acyclic nets (SAFC nets) or even occurrence nets and policies are defined by a constant deactivation pattern. We obtain complexity-theoretic results for such cases via a close connection to Bayesian networks, in particular we show that for SAFC nets the question whether there is a policy guaranteeing a reward above a certain threshold is $$\textsf{NP}^\textsf{PP}$$ -complete. We also introduce a partial-order procedure which uses an SMT solver to address this problem.

Taking Complete Finite Prefixes to High Level, Symbolically

Abstract: Unfoldings are a well known partial-order semantics of P/T Petri nets that can be applied to various model checking or verification problems. For high-level Petri nets, the so-called symbolic unfolding generalizes this notion. A complete finite prefix of the unfolding of a P/T Petri net contains all information to verify, e.g., reachability of markings. We unite these two concepts and define complete finite prefixes of the symbolic unfolding of high-level Petri nets. For a class of safe high-level Petri nets, we generalize the well-known algorithm by Esparza et al. for constructing small such prefixes. Additionally, we identify a more general class of nets with infinitely many reachable markings, for which an approach with an adapted cut-off criterion extends the complete prefix methodology, in the sense that the original algorithm cannot be applied to the P/T net represented by a high-level net.

Petri-Net-Based Model Checking for Privacy-Critical Multiagent Systems

Abstract: Computation tree logic of knowledge (CTLK) can be used to specify many properties related to privacy of multiagent systems (MASs). Our previous work defined knowledge-oriented Petri nets (KPNs) to formally describe both the interacting/collaborating process of multiagents and their epistemic evolutions. Our KPN-based verification of CTLK required to obtain all reachable states, the transition relation of all states, and the equivalence relations of all states with respect to knowledge, which resulted in a serious state explosion problem and thus only fit some small-scale systems. This article adopts reduced ordered binary decision diagram (ROBDD) to deal with this problem. Especially, the ROBDD technique is used to encode and store all reachable states but not to encode and store any transition relation or equivalence relation. However, when verifying a CTLK formula, the transition relation and equivalence relation of some states must be known. To solve this problem, we design the related algorithms to compute only those required relations and prove their correctness. We design the related model checking algorithms and develop a tool. A number of experiments are done by using a famous benchmark about the privacy problem of MAS: the dining cryptographers protocol (DCP), and the results illustrate the advantages of our methods. For example, our tool running with a general PC spends less than 14 h to verify the DCP with 1200 cryptographers, which involves about 101080 states and two CTLK formulas with more than 6000 atomic propositions and more than 3600 operators. This represents a significant advance in the field of model checking.