Reachability Tree Research Articles

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.

Analysis of Signed Petri net

In this paper, the behavioural properties of Signed Petri net (SPN) are given along with the two techniques: reachability tree and matrix equations to analyse the SPN. An actual case scenario of a restaurant model is given and analysed using the techniques mentioned in the paper. The benefits of using an SPN to model the restaurant system rather than using Petri net are also given.

An Integrated Method for Solving the Two-Sided Assembly Line Balancing Problems

The problems of parallel two-sided assembly lines balancing (TALB) are nonpolynomial (NP-hard) problems that belong to the field of combinatorial optimization. In most cases, these problems can be resolved with algorithms. To this end, this research adopts colored Petri nets (CPN) and a minimal spanning tree (MST) to address the type I problems for two-sided assembly line balancing problem (TALBP). This research proposes a three-step method. In the first step, a reachability tree chart is built to show potential tasks order assignment. In the second step, the color firing mechanism is employed to determine if the colored conditions are satisfied. The final step uses a MST and a least-time prioritized rule to select the proper tasks for assignment efficiently. These steps will be performed repetitively until all the tasks are properly assigned. Since the whole assignment process is based on graphical and paradigm methods, it can be adopted to lessen the burdens of complex calculations. The end of the research will present the comparison results between the examples from the literature and the method proposed in this research. As indicated by the research findings, employing CPN and an MST for TALBPs (type-I) can obtain even favorable and consistent results over legacy methods, and further optimize workstation design and enhance balance efficiency.

Modeling and Verification of Aircraft Takeoff Through Novel Quantum Nets

The formal modeling and verification of aircraft takeoff is a challenge because it is a complex safety-critical operation. The task of aircraft takeoff is distributed amongst various computer-based controllers, however, with the growing malicious threats a secure communication between aircraft and controllers becomes highly important. This research serves as a starting point for integration of BB84 quantum protocol with petri nets for secure modeling and verification of takeoff procedure. The integrated model combines the BB84 quantum cryptographic protocol with powerful verification tool support offered by petri nets. To model certain important properties of BB84, a new variant of petri nets coined as Quantum Nets are proposed by defining their mathematical foundations and overall system dynamics, furthermore, some important system properties are also abstractly defined. The proposed Quantum Nets are then applied for modeling of aircraft takeoff process by defining three quantum nets: namely aircraft, runway controller and gate controller. For authentication between quantum nets, the use of external places and transitions is demonstrated to describe the encryption-decryption process of qubits stream. Finally, the developed takeoff quantum network is verified through simulation offered by colored petri-net (CPN) Tools. Moreover, reachability tree (RT) analysis is also performed to have greater confidence in feasibility and correctness of the proposed aircraft takeoff model through the Quantum Nets.

Modeling a New AQM Model for Internet Chaotic Behavior Using Petri Nets

Formal modeling is considered one of the fundamental phases in the design of network algorithms, including Active Queue Management (AQM) schemes. This article focuses on modeling with Petri nets (PNs) a new scheme of AQM. This innovative AQM is based on a discrete dynamical model of random early detection (RED) for controlling bifurcations and chaos in Internet congestion control. It incorporates new parameters (α,β) that make possible better stability control over oscillations of an average queue length (AQL) at the router. The PN is validated through the matrix equation approach, reachability tree, and invariant analysis. The correctness is validated through the key properties of reachability, boundedness, reversibility, deadlock, and liveness.

Toward Practical Weakly Hard Real-Time Systems: A Job-Class-Level Scheduling Approach

Recent applications of the Internet of Things and cyber–physical systems require the integration of many sensing and control tasks into resource-constrained embedded devices. Such tasks can often tolerate a bounded number of timing violations. The concept of weakly hard real-time systems can effectively improve resource efficiency without sacrificing system safety. However, the existing studies have limitations on their practical use due to the restrictions imposed on the task timing behavior, high analysis complexity, and the lack of multicore support. In this article, we propose a new job-class-level fixed-priority preemptive scheduler and its schedulability analysis framework for weakly hard real-time tasks. Our proposed scheduler employs the meet-oriented classification of jobs of a task in order to reduce the worst-case temporal interference imposed on other tasks. Under this approach, each job is associated with a “job-class” that is determined by the number of deadlines previously met (with a bounded number of consecutively missed deadlines). This approach allows decomposing the complex weakly hard schedulability problem into two subproblems that are easier to solve: 1) analyzing the response time of a job with each job-class, which can be done by an extension of the existing task-level analysis and 2) finding possible job-class patterns, which can be modeled as a simple reachability tree. We also present a semipartitioned task allocation method for multicore platforms, which enhances the schedulability of weakly hard tasks under the proposed scheduling framework. Experimental results indicate that our scheduler outperforms the prior work in terms of task schedulability and analysis time complexity. We have also implemented a prototype of a job-class-level scheduler in the Linux kernel running on Raspberry Pi with acceptably small-runtime overhead.

Axiomatization, Computability and Stability for Discrete Event Time Algorithms

This work proposes a formalization of discrete event time algorithms. A Church type thesis, its proof, and the notion of stability for discrete event time algorithms are presented. The Church thesis for discrete algorithms motivates us to consider the Church thesis for the case when we are dealing with discrete event time algorithms. The notions of discrete event time algorithm and discrete event time dynamical system are postulated to be equivalent. The stability concept for discrete event time algorithms is defined. The stability analysis presentation starts concentrating in discrete event time algorithms, i.e., discrete event time dynamical systems, whose Petri net model is described by difference equations, and continues considering Lyapunov energy function sin terms of the logical structures of the vocabulary. A stability analysis based on the reachability tree of the Petri net model is also discussed.

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.

Reachability problem in non-uniform cellular automata

This paper deals with the CREP (Configuration REachability Problem) for non-uniform cellular automata (CAs). The cells of non-uniform CAs, we have considered here, can use different Wolfram’s rules to generate their next states. We report an algorithm which decides whether or not a configuration of a given (non-uniform) cellular automaton is reachable from another configuration. We develop a characterization tool, named Reachability tree, is used to develop theories and the decision algorithm for the CREP. Though the worst case complexity of the algorithm is exponential in time and space, but the average performance is very good.

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.

Divide Large Scale Fuzzy Petri Net into Equivalent Sub-FPNs using Reachability Tree

As real systems become larger and complicated, the number of nodes of fuzzy Petri net (FPN) also increases rapidly, it further indicated that the algorithm complexity of related application using FPN grows up sharply. To solve this state-explosion issue, a decomposition algorithm using reachability tree is proposed to divide a large-scale FPN model into a series of sub-FPNs in this paper. Comparison experiments show that all obtained sub-FPNs do not destroy internal reasoning paths of the original large-scale FPN and represent a complete reasoning path from input place(s) to the output place.

Modelling race conditions in multithreading programs in terms of Petri nets

The article describes an approach to detecting race conditions with respect to a variable in a program, based on the automatic construction of a program model in terms of Petri nets. A simple sample program having race condition is given and the analysis of its behavior shows four different results of execution due to the race condition. Compositional model of the program in terms of Petri nets is presented that illustrates the process of automatic model generation. Notions of program control flow, global area and environment models are introduced. Reduction rules for the control flow model are defined and compositional representation of reduced program model is described. The formula for detection of race conditions is defined as: for a race conditions to occur, it is necessary that between the moment a variable is read by a thread and the moment this thread writes to a variable, an event of writing to the variable by another thread may occur. Compositional model of global variable capable of intercepting writing to variable events that lead to race condition is presented. By synchronizing the program control flow model with the global variable model it is possible to detect race conditions by building the reachability tree and checking for the presence of error event in it.

Hybrid Dynamic Probability-Based Modeling Technique for Rolling Stock Failure Analysis

The purpose of this study is to propose a novel hybrid dynamic probability-based failure analysis technique consisting of dynamic Bayesian discretization (DBD) and stochastic Petri nets (SPNs) for railway rolling stock (RS) failure analysis. Performing failure analysis and diagnoses for integrated RS subsystems is challenging and can lead to operational delays affecting fleet reliability and availability. This paper presents an integrated feature of updative adaptation using DBD methods to analyze prior continuous and discrete probability data—by means of evidence-based propagation to ascertain posterior faulty component states and simultaneously allowing for rapid failure notification, detection, and isolation of multiple RS subsystems using the reachability tree characteristics of SPNs. Unlike other dynamic probability methods, the DBD-SPN hybrid model presented here reduces computational time and enhances convergence accuracy using the Kullback–Leibler measure, sequential event analysis, and stable and low-entropy-error characteristics. In an extensive UK-based RS case study, it was observed that this approach is suitable for rapid failure notification, detection, and isolation of traction door interlock failure. It is also believed that the current study represents a useful contribution to the research and technology of hybrid DBD and SPNs for the failure analysis of a system consisting of multiple subsystems, since its application makes the difference between being able to evaluate realistically common cause and sequential failure analyses of complex systems.

Graph simulation on large scale temporal graphs

Graph simulation is one of the most important queries in graph pattern matching, and it is being increasingly used in various applications, e.g., protein interaction networks, software plagiarism detection. Most previous studies mainly focused on the simulation problem on static graphs, which neglected the temporal factors in daily life. In this paper, we investigate a novel problem, namely, temporal bounded simulation on temporal graphs. Specifically, given a pattern query Q with bound constraint and a temporal graph G, we aim to find out the matches Q(G) of Q on G, such that Q(G) satisfies temporal reachability and bound constraint of Q. To tackle this problem efficiently, we propose a simulation matching framework, which consists of three phases, namely pattern segmentation, temporal bounded simulation of pattern segments, and result integration. We first divide the Q into simple small subgraphs (segments). We construct a temporal reachable tree of G. Based on the tree, we propose a basic matching algorithm for the pattern segments. Furthermore, we propose two optimization algorithms by leveraging multi-layer partition strategy to accelerate query processing. After that, we integrate the simulation results obtained from the pattern segments according to the constraints of temporal reachability and the bound of query pattern. Finally, we use real temporal and synthetic datasets to empirically verify the efficiency and effectiveness of our solutions for temporal bounded simulation matching.

Existence of Forbidden Digraphs for Crisp Boolean Petri Nets

Boolean Petri net (BPN) and Crisp Boolean Petri net (CBPN) is a well-studied graph model since 2010 which has several applications in mathematical modeling of complex or tricky networks. Modeling any network with Petri net which can generate binary numbers as marking vectors in its reachability tree is still has much uses. In CBPN with a minimum number of transition and minimum number of steps of reachability tree, minimal execution time to run the machine has not been noted till date, thus it’s necessary to sort out this problem. Possibly it may occur due to some forbidden structure which hinders any 1-safe Petri net to be a CBPN. In this paper, we present some forbidden digraphs whose presence interrupts the generation of binary n-vectors exactly once. Any 1-safe Petri net is not a CBPN if it contains any of the subnet induced to the four forbidden structures discussed in this paper.

MODELING THE DYNAMIC INTERACTION PROCESSES USING OF FUZZY TIMED PETRI NETS OF TYPE VF

The dynamic interacting processes modeling is examined in the article, which shows the complex objects operation in the condition of uncertainty. A formalism intended for the development and analysis models of complex parallel and distributed systems is proposed. It is based on the mathematical apparatus of the fuzzy timed Petri nets (FTPN) of type Vf, representing generalized FTPN of type Vf , combining deterministic and non-deterministic character. The algorithm for the functioning FTPN of type Vf is developed. The proposed algorithm provides a solution to the problem of the triggering solvability of transitions occurring in conflict states, the imposition of a fuzzy structure on the network marking with fuzzy composition laws that determine the values the degrees of belonging the input and output transition positions. The model of parallel functioning processing devices is presented in the FTPN form of type Vf. An approach is proposed for modeling dynamic interacting processes based on the matrix theory of Petri nets, that provides an effective form of structure representation, model state dynamics, the space of achievable states, and triggering transitions sequence in the form of vectors and matrices set. On the example of the production model of mechanical processing, it is shown that the accepted triggering transitions rules fully show the functioning FTPN process of type Vf. As a result of the simulation, the reachability tree is obtained as a sequence of matrices.

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.

Verifying the Safety of Aviation Software Based on Extended Colored Petri Net

At present, various functions of aircraft have become more and more dependent on airborne software system, and the structure of modern airborne software is extremely complex. Engineers need to eliminate the situation in which the safety performance of the entirety reduces caused by mutual influence among components. This paper presents a comprehensive method with high efficiency for safety verification of airborne software system, in order to ensure the system meet safety requirement of airworthiness standard at the design stage. Safety Verification Colored Petri Net (SVCPN) for software safety verification is firstly proposed, and then the mapping transformation rules from Block Definition Diagram (BDD) of System Modeling Language (SysML) to SVCPN are proposed to achieve the accurately formal description of software system. Traversing all delivery paths of safety level transfer based on the Reachable Tree Diagram, to detect the components that do not meet the safety requirement of airworthiness standard. Based on the disambiguation algorithm, the fundamental components that cause safety problem are found out through the establishment of antinet to achieve the safety level reassign, ensuring the safety performance of the whole system. Finally, the case study and the comparison and analysis are applied to show the feasibility and superiority of our method.

Petri Net Recommender System for Generating of Perfect Binary Tree

In modeling a discrete event driven system, Petri net recommender systems can play a very important role in describing the structural and behavioral properties of complex and tricky networks. The finite and infinite perfect binary tree forms a predictive model which can map the input information to output information based on the inputs' attributes. A perfect binary tree can be used for three types of recommender systems such as: collaborative filtering, a content-based approach, and a hybrid approach. In this article, the authors show the existence of a Petri net whose reachability tree is a Perfect Infinite Binary Tree (PIBT).

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

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