Reachability Graph Research Articles

A Mathematical Model Using Time Elements for Timed-Arc Petri Nets

In this work, a novel graphical and mathematical model is introduced for Timed-Arc Petri Nets. In this model, operation durations related to events are associated with arcs as firing durations. Time elements are introduced on the arcs of the model for monitoring tokens in operation durations (firing durations) of firing processes (events). In the model, the status of the system is represented as states consisting of remaining time vector for indicating the status of time elements and marking vector for indicating the status of places' state information. This feature of states allows to obtain reachability set enhanced with the time information and the timed reachability graph of the system. In this work, behavioral properties of the proposed model are also defined. Moreover, via considering real world systems, behavioral properties are analyzed and performance of the model is compared with Stretched Petri Nets which is a recently proposed type of Timed Petri Nets.

ProSy: API-Based Synthesis with Probabilistic Model

Program synthesis is an exciting topic that desires to generate programs satisfying user intent automatically. But in most cases, only small programs for simple or domain-specific tasks can be synthesized. The major obstacle of synthesis lies in the huge search space. A common practice in addressing this problem is using a domain-specific language, while many approaches still wish to synthesize programs in general programming languages. With the rapid growth of reusable libraries, component-based synthesis provides a promising way, such as synthesizing Java programs which are only composed of APIs (application programming interfaces). However, the efficiency of searching for proper solutions for complex tasks is still a challenge. Given an unfamiliar programming task, programmers would search for API usage knowledge from various coding resources to reduce the search space. Considering this, we propose a novel approach named ProSy to synthesize API-based programs in Java. The key novelty is to retrieve related knowledge from Javadoc and Stack Overflow and then construct a probabilistic reachability graph. It assigns higher probabilities to APIs that are more likely to be used in implementing the given task. In the synthesis process, the program sketch with a higher probability will be considered first; thus, the number of explored reachable paths would be decreased. Some extension and optimization strategies are further studied in the paper. We implement our approach and conduct several experiments on it. We compare ProSy with SyPet and other state-of-the-art API-based synthesis approaches. The experimental results show that ProSy reduces the synthesis time of SyPet by up to 80%.

Scheduling Robotic Cellular Manufacturing Systems With Timed Petri Net, A* Search, and Admissible Heuristic Function

System scheduling is a decision-making process that plays an important role in improving the performance of robotic cellular manufacturing (RCM) systems. Timed Petri nets (PNs) are a formalism suitable for graphically and concisely modeling such systems and obtaining their reachable state graphs. Within their reachability graphs, timed PNs’ evolution and intelligent search algorithms can be combined to find an efficient operation sequence from an initial state to a goal one for the underlying systems of the nets. To schedule RCM systems, this work proposes an A* search with a new heuristic function based on timed PNs. When compared with related approaches, the proposed one can deal with token remaining time, weighted arcs, and multiple resource copies commonly seen in the PN models of RCM systems. The admissibility of the proposed heuristic function is proved. Finally, experimental results are given to show the effectiveness and efficiency of the proposed method and heuristic function. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Robotic cellular manufacturing (RCM) systems are among the most common and complicated discrete-event dynamic systems, which provide a great number of choices of resources and processing routes to allow high system productivity. Timed Petri nets (PNs) and intelligent search algorithms on their reachability graphs are ideal tools to handle the RCM scheduling problem. This work proposes an A* search method based on the evolutions of timed PNs to optimally schedule RCM systems. The proposed method can deal with token remaining time, weighted arcs, and multiple resource copies often encountered in the PN models of RCM systems.

On Hierarchical Construction of the State Space of an Automated Manufacturing System Modeled With Petri Nets

State space techniques are one of the main approaches deployed for the analysis of concurrent systems. However, state space construction is stalled by a common phenomenon called the state explosion problem which makes it a tough task or even impossible when the state space computation demands prohibitive cost (time and memory). We limit general resource allocation systems (RASs) to a certain class whose state space can be hierarchically constructed, yet it comprises various enough types of real-world discrete event systems, such as automated manufacturing systems. This paper focuses on a class of RASs modeled with Petri nets (PNs), where, through pure algebraic operations, a novel method to compute the state spaces is proposed, which is motivated by the superposition property. Given a PN model of a system and a target resource configuration, we first propose a special initial marking called the initial basis marking and compute the corresponding reachability graph. Then, we increase the capacity of the resource places in an incremental way and generate the reachability graphs by taking advantage of the PN structure and the previously computed reachability graph until the capacity function of resources reaches the target resource configuration. A complete enumeration of reachable states can be obtained by a recursive scheme. Experimental studies also demonstrate the efficiency of the proposed approach in terms of computational cost and its high-potential to cope with the state-explosion problem.

A Dynamic Data Slice Approach to the Vulnerability Analysis of E-Commerce Systems

The e-commerce business process net (EBPN) is a novel formal model for describing an e-commerce system and its interactive parts, such as shoppers, merchants, and the third-party payment platforms. Vulnerability analysis has a great impact on the trustworthiness of EBPN, which is an issue stemming from data inconsistency problems. Data inconsistency problems affect the consistency of the EBPN transaction analysis. The underlying causes of inconsistent data are closely related to concurrent operations, such as control flow and data flow. However, most of the existing detection methods have difficulties characterizing the vulnerabilities and interactions of control and data flows. In this paper, we propose a new method based on the dynamic data slice (DDS) that considers both transaction consistency and data state consistency. First, by analyzing control flow characteristics of EBPN, we obtain the dynamic slice. This dynamic slice is based on all paths of the EBPN reachability graph. Second, we perform the data inconsistency analysis by considering both transaction consistency and data-state consistency. Based on these, we construct a DDS to characterize the behavioral logic and the data-dependence information. The DDS acquires the dynamic data firing sequence. Based on that sequence and a given data marking, we can construct the DDSs for several types of EBPNs. Constructing the DDS can be completed in polynomial time. The DDS is designed to characterize the behavioral logic and data-dependence information. Based on these, we design a method to judge the data constraints. This method satisfies the EBPN need for transaction consistency by considering both the control and data states. In addition, according to the data-dependence information, we can lock the vulnerable regions caused by abnormal trading data in the system. Finally, we give a method to compute the vulnerability level.

On the Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency

Modeling of real-world systems with Petri nets allows to benefit from their generic concepts of parallelism, synchronisation and conflict, and obtain a concise yet expressive system representation. Algorithms for synthesis of a net from a sequential specification enable the well-developed theory of Petri nets to be applied for the system analysis through a net model. The problem of $\tau$-synthesis consists in deciding whether a given directed labeled graph $A$ is isomorphic to the reachability graph of a Boolean Petri net $N$ of type $\tau$. In case of a positive decision, $N$ should be constructed. For many Boolean types of nets, the problem is NP-complete. This paper deals with a special variant of $\tau$-synthesis that imposes restrictions for the target net $N$: we investigate dependency $d$-restricted tau-synthesis (DR$\tau$S) where each place of $N$ can influence and be influenced by at most d transitions. For a type $\tau$, if tau-synthesis is NP-complete then DR$\tau$S is also NP-complete. In this paper, we show that DR$\tau$S parameterized by $d$ is in XP. Furthermore, we prove that it is W[2]-hard, for many Boolean types that allow unconditional interactions set and reset.

Refinement-Based Hierarchical Modeling and Correctness Verification of Cross-Organization Collaborative Emergency Response Processes

When an emergency occurs, one of the important challenges is how to form an effective and timely response. An emergency disposal plan is usually organized as a series of emergency response processes manipulated by one emergency command center and several subordinate emergency organizations. Moreover, these subordinate organizations are usually geographically dispersed and need to collaborate with each other. In this case, designing and verifying such cross-organizational collaborative emergency response processes are complicated and time-consuming. To address this problem, we propose a hierarchical modeling and correctness verification approach. A general framework for hierarchical modeling and correctness verification of such processes is first introduced. Then, a top-level model and two kinds of bottom-level models (complex and simple bottom-level ones) are proposed to model such processes and collaboration patterns from different abstraction levels. Next, Petri net refinement operation is adopted to refine the top-level model by using its corresponding bottom-level models to obtain the refined model. Finally, the correctness of the refined model is verified based on reachability graph. A typical running case of cross-organization collaborative fire emergency response processes is given to validate our proposed method.

Development of a New Strategy to Extract Dangerous Scenarios from Petrochemical Industry Installation

The use of Petri net reachability graph remains one of the most popular methods to extract critical scenarios that lead the system to a dangerous state. However, in complex systems, explosion states space and confusion between causality and precedence relationship between events are the two major limits making reachability graph inefficient to perform such analysis. In the last decade, the first limitation was tackled by an approach that uses the Petri net structure. It considers only the last normal state and ignores the rest of the network. Nevertheless, no research work appears in the literature, to consider the second limitation. In this sense, this paper proposes a novel approach based on Petri net and linear logic, to overcome the two limits. To prove the effectiveness of this proposal, the approach was applied on a petrochemical installation consisting of a cooling flammable fluids storage bins system. The obtained results are compared with the two existing approaches, the first using reachability graph and the second using the Petri net structure. The new proposed approach has shown higher performances compared to the previously mentioned methods.

Control Design for Bounded Partially Controlled TPNs Using Timed Extended Reachability Graphs and MDP

This paper is about the design of control sequences for discrete event systems (DESs) modeled with bounded partially controlled timed Petri nets (PC-TPNs) including a set of temporal specifications that correspond to minimal firing durations. Petri nets are well-known mathematical and graphical models that are widely used to describe distributed DESs, including choices, synchronizations, and parallelisms. The domains of application include but are not restricted to manufacturing systems, computer science, and transportation networks. Including the time in the model is important to consider many control problems. This paper is more particularly concerned with control issues in timed context and uncertain environments when unexpected events occur and when control errors disturb the system behavior from the planned policy decided by the controller. To deal with such uncertainties, we propose first to build a timed extended reachability graph (TERG) that includes the time specifications when the PC-TPN behaves with an earliest firing policy. Then, the optimal paths in TERG are found by using an approach based on discrete time Markov decision processes under discounted criterion. Several simulations illustrate the benefit of our method from the computational point of view, in particular for uncertain environments.

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.

A deadlock control algorithm using control transitions for flexible manufacturing systems modelling with Petri nets

ABSTRACTThis paper develops a concept of control transition equation (CTE) and the corresponding deadlock control algorithm (DCA) using control transitions (CTs) to eliminate deadlocks in Petri nets. By analysing the reachability graph (RG) of an original net with deadlocks, this DCA firstly find all deadlock markings (DMs) and then the corresponding CTs are solved on the basis of the proposed CTE. Secondly, a linear programming problem that can minimise the number of CTs is applied to these CTs. In addition, in order to furtherly simplify the structure of these CTs, the reconstruction of the necessarily added CTs is performed by a circulating sequence number method. Finally, a live controlled system with the simpler structure is obtained by adding the desired CTs to , which can reach the same number of states as the original plant model , i. e. live maximally reachable number. The proposed DCA is different from deadlock prevention policies using control places (CPs) in most existing literature and whose correctness and efficiency are verified via the theoretical analysis and the relevant several examples.

Marking Estimation in Petri Nets Using Hierarchical Basis Reachability Graphs

In this article, we propose a two-layer-structure called hierarchical basis reachability graph that is useful for marking estimation in Petri nets that contain unobservable transitions. On the basis of a hierarchical partition of the set of observable transitions, a hierarchical basis reachability graph consists of a primary and a secondary structures. Simulation shows that the time-consumption to compute a hierarchical basis reachability graph is in general much less than that of computing the corresponding basis reachability graph. A marking estimation algorithm is proposed by the structural analysis of the corresponding hierarchical basis reachability graph.

Current-state opacity modelling and verification in partially observed Petri nets

System opacity is a widely studied security notion, implying that a secret behaviour of a given system cannot be seen or assessed by an external observer based on the system evolution. This work deals with the problem of current-state opacity formulation and verification in the context of discrete event systems modelled with partially observed Petri nets (POPNs) (i.e., Petri nets containing place sensors that measure the number of tokens in observable places and event sensors that indicate the firing of observable transitions). A Petri net system is recognized as current-state opaque if the current-state estimate is never entirely contained in the set of secret states. In this regard, we introduce the notion of discernible markings to design a reduced state estimator called a discernible reachability graph, and then come up with formal modelling of current-state opacity in POPN systems. The main idea of the proposed approach consists in proving that if a system is current-state opaque, its current-state estimate, possibly established by an intruder, contains at least one non-secret state. We exploit the mathematical feasibility to formulate this concept by defining and solving an integer linear programming problem with respect to a given secret and an observation sequence collected from sensors. In the light of the proposed modelling, necessary and sufficient conditions are proposed for opacity verification, and examples are given to expose the results.

On the Opportunistic Topology of Taxi Networks in Urban Mobility Environment

Understanding and characterizing the connectivity of vehicular networks has become increasingly important because of their wide applications and fast development. To address the dynamical links in vehicular networks, time-varying graph (TVG) is one of the most important models. Nowadays, due to the fact that lots of vehicular applications can tolerate a small amount of latency in communication, opportunistic reachability graph (ORG) characterizes the connectivity better by introducing delay tolerance to the model. However, people still do not have a high-level summarization, i.e., the topology, of the vehicular network on how nodes are clustered and isolated. In this paper, based on ORG model, we analyze the opportunistic topology of taxi networks in urban mobility environment by mainly focusing on the number, location and evolution of connected components and the size of the largest components to reveal the unique properties of the taxi networks instead of just links and hops. Our analysis is based on the real taxi traces of big cities and reflects the real urban mobility environment. We find that the opportunistic topology of the networks with delay tolerance is substantially different from the instantaneous topology without considering the delay. Moreover, we unveil the fundamental relationships and trade-offs between the dynamical topology and the key network parameters related to mobility, e.g., delay tolerance, transmission distance, etc. To the best of our knowledge, our study is the first work to reveal the characteristics of opportunistic topology models in the large-scale urban mobility environment with real traces.

Development of Collaborative Business Processes: A Correctness Enforcement Approach

Collaborative business processes gather a set of business processes with complementary competencies and knowledge to cooperate to achieve more business successes. To ensure their successful implementation, correctness is a key issue that needs to be addressed during their development. To this end, a novel correctness enforcement approach is proposed to support the development of collaborative business processes. In this approach, we first give an algorithm to check the correctness of an original process specified by Petri nets. Then, we prune its reachability graph to obtain its core in case of partially correct, which is a reduced reachability graph that doesn't cover invalid states. Finally, we generate a set of controllers from the core using coordination mapping (i.e., inserting some coordination activities into controllers), and then an enforced process is built by the composition of the original process and the controllers. Our approach is implemented as an analysis module called cetool in the PIPE (Platform Independent Petri Net Editor) and it is validated on a set of real-world cases. The results show the effectiveness and efficiency of the proposed approach.

Online Fault Diagnosis of Labeled Petri Nets Based on Reachability Graphs and Topological Sorting

In this article, we propose an online fault diagnosis approach for labeled Petri nets. When observing an event, the reachability graph of the unobservable subnet (i.e., the net consisting of all places, unobservable transitions, and the attached arcs between them) is first constructed and then a topological sort (i.e., an ordered list of markings) of the reachability graph is obtained. According to the topological sort, each marking in the reachability graph is associated with a vector that contains diagnosis information to form a new graph, called a diagnosis graph. Based on the diagnosis graph constructed for each observed event, an online algorithm is developed to perform diagnosis. When the considered Petri net systems have a small amount of unobservable transitions, the proposed approach enjoys a high computational efficiency compared with the existing ones using integer linear programming.

Vulnerability Evaluation Method for E-Commerce Transaction Systems With Unobservable Transitions

E-commerce transaction systems have become an important factor in trading activities. However, e-commerce systems are still undergoing development. Unobservable actions and attacks on systems are frequent problems that increase the vulnerability of e-commerce systems. Most existing approaches to addressing these issues cannot describe or analyze the overall structure of a local specification and unobservable actions well. The vulnerable e-commerce transaction net (VET-net) is a useful model for describing the unobservable actions, online transactions and third-party payment platforms of e-commerce systems. Based on a VET-net, we focus on the detection and evaluation of e-commerce transaction systems to attacks. We propose the concept of vulnerable transitions, which include not only vulnerable actions but also unobservable transitions. Then, we use an improved slice method to locate the vulnerable transitions. For these vulnerable transitions, we propose a vulnerable transition evaluation method based on a hidden Markov model along with a reachability graph (HMM-RG). The HMM-RG uses hidden Markov models (HMMs) to approximate the state reachability graph of a VET-net. By calculating the firing probability, the HMM-RG can evaluate the vulnerability degree of malicious states. We use a real-world case to show our method’s effectiveness and reasonability.

Partial Reachability Graph Analysis of Petri Nets for Flexible Manufacturing Systems

Petri nets are an important and popular tool to model and analyze deadlocks in flexible manufacturing systems. The state space of a Petri net model can be divided into two disjoint parts: a live-zone and a dead-zone. Reachability graph analysis plays an important role in the modeling and control of Petri nets. Most existing studies have to fully enumerate the reachable markings of a Petri net to obtain the first-met bad markings (FBMs), which exacerbates the computational overheads. In this paper, a computationally efficient method to find dead markings in Petri nets is presented. We first introduce an algorithm to find dead markings by solving an integer linear programming problem. Then, the set of markings in the dead-zone is calculated, including the set of dead markings and the set of bad markings. Then we can find all the FBMs. By using a vector covering approach, the minimal covered set of FBMs is computed. The proposed approach can obtain the dead markings and FBMs by searching only a part of a reachability graph. Finally, examples are provided to demonstrate the proposed method.

Verification of Infinite-step Opacity Using Labeled Petri Nets

Abstract Opacity is an important information secure property. A system is said to be infinite-step opaque if the intruder is never able to ascertain that the system is or has been in a secret state at some time, based on its observation of the system evolution. This work aims to verify infinite-step opacity of discrete event systems modeled with labeled Petri nets. Based on the notion of basis reachability graph, a new structure called basis two-way observer is proposed to check infinite-step opacity of a bounded system, which is shown to be more efficient than the standard method based on the reachability graph.

On Multi-Step Look-Ahead Deadlock Prediction for Automated Manufacturing Systems Based on Petri Nets

Automated manufacturing systems (AMSs) are prone to be in deadlock states when resource allocation is unreasonable. Reasonably allocating system resources to achieve deadlock control is a primary task of the design of an AMS. This paper proposes a multi-step look-ahead deadlock prediction method to obtain an optimal deadlock avoidance policy for a class Petri nets, without calculating a complete reachability graph. As the reachability graph of a large-scale Petri net model is usually large, the analysis and calculation process to obtain the optimal deadlock avoidance policy is complicated. We first simplify a Petri net model by using the existing Petri net reduction methods or removing non-shared resources to simplify the model structure. Then we calculate the dead zone markings of the simplified model through a reverse generation method, and finally develop a multi-step look-ahead deadlock prediction method to obtain an optimal deadlock avoidance policy for a class of AMSs.