Reachability Tree Research Articles

An Automatic Algorithm to Generate a Reachability Tree for Large-Scale Fuzzy Petri Net by And/Or Graph

Fuzzy Petri net (FPN) is widely used to repre sent, model and analyse knowledge-based systems (KBSs). Meanwhile, a reachability tree is an important tool to fully represent the flow relationship of FPN and is widely applied to implement inference in industrial areas. However, the traditional reachability ignores recording the dependence relationships (‘and/or’ relationship) among the places in the neighbouring layers. This paper develops a modified reachability tree based on an and/or graph and presents a three-phase generation algorithm to model the reachability tree for the corresponding FPN automatically via fuzzy production rules (FPRs). Four cases are used to verify the correctness and feasibility of the proposed algorithm from different viewpoints, such as general FPRs, FPRs with a condition-sharing situation, FPRs with a conclusion-sharing situation, and FPRs with multi-conclusions. Simulation results reveal that the proposed approach has the ability to automatically generate the reachability tree for the corresponding FPN correctly.

Container survey of Aedes breeding sites in a Rural Health Training Centre campus in South West of Delhi

Introduction: Aedes mosquito is one of the mosquito species that are most studied because of their potential as vector of dengue and such information can be used to design an effective control measures for campuses. Objective: To study the Aedes breeding sites using the container survey method and to recommend control measures in controlling the Aedes breeding sites. Methodology: The container survey study was carried out in the campus of a Rural health training centre campus which spreads over 30 acres with rich plantation and trees all around. All natural and artificial containers in the vicinity of the campus and reachable tree holes were examined during the survey. The number, type and water condition of containers that may serve as potential breeding sites were examined and recorded. Larvae collection was carried out indoors (interior of the building) and outdoors (outside of the building but confined to its immediate vicinity within the campus limits) by using pre-designed questionnaire. Larva identification was done by motility Pattern. Result: The survey included study of 359 common breeding sites such as cooler, Plastic storage containers, unused plastic waste, cement tanks, syntax tanks, earthen pots, plant pots, solid waste and others in 13 sites and 3 out of the 359 containers contained Aedes larvae. Conclusion: The RHTC management should Practise Integrated Vector Management (IVM) recommended by WHO in the campus. An IVM approach takes into account the available health infrastructure and resources, and integrates all available and effective measures, whether chemical, biological or environmental.

Failures in discrete-event systems and dealing with them by means of Petri nets

An approach based on Petri nets pointing to the manner how to deal with failures in discrete-event systems is presented. It uses the reachability tree and/or reachability graph of the Petri net-based model of the real system as well as the synthesis of a supervisor to remove the possible deadlock(s). To illustrate the applicability of the approach to the detection and recovery of failures in DES modelled by Petri nets the case study on a railroad crossing is introduced.

Lean Reachability Tree for Unbounded Petri Nets

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

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

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

Graphical Modeling and Analysis Software for State Space-Based Optimization of Discrete Event Systems

In view of the particular research objectives rather than the system’s characteristics, almost all systems can be discretized regardless of original continuous or discrete pattern. Modeling oriented to discrete-event system (DES) represents the dynamics of a system as a series of discrete events that perform changes in the state of the system, constituting the state space which supports the further analysis for scheduling and optimization. In this paper, the graphical modeling and analysis software (GMAS) as a platform for modeling DES is introduced with the basic notions and a general perspective on the systems approach. It clearly provides the graphic modeling and analysis interface. Besides, the system evolution process is recorded and represented by state space, transforming the optimization problem into a search-based issue in the reachability tree of finding the optimal or near-optimal sequence of function component activations from some initial state to the goal state. To validate its efficacy and practicability, a causal encounter model of traffic collision avoidance system operations is proposed in the GMAS formalism. The model has been proved to not only provide a better comprehension of the potential collision occurrences for risk assessment by representing the cause–effect relationship of each action but also aid the crews in the involved aircraft to make a cooperative and optimal option.

Scalable Verification of Networks With Packet Transformers Using Atomic Predicates

Packet transformers are widely used in ISPs, datacenter infrastructures, and layer-2 networks. Existing network verification tools do not scale to large networks with transformers (e.g., MPLS, IP-in-IP, and NAT). Toward scalable verification, we conceived a novel packet equivalence relation. For networks with packet transformers, we first present a formal definition of the packet equivalence relation. Our transformer model is general, including most transformers used in real networks. We also present a new definition of atomic predicates that specify the coarsest equivalence classes of packets in the packet space. We designed an algorithm for computing these atomic predicates. We built a verifier, named A tomic P redicates for T ransformers, and evaluated its performance using four network data sets with MPLS tunnels, IP-in-IP tunnels, and NATs. For a provider cone data set with 11.6 million forwarding rules, 92 routers, 1920 duplex ports, and 40 MPLS tunnels which use 170 transformers, APT used only 0.065 s, on average, to compute the reachability tree from a source port to all other ports for all packets and perform loop detection as well. For the Stanford and Internet2 data sets with NATs, APT is faster than HSA (Hassel in C implementation) by two to three orders of magnitude. By working with atomic predicates instead of individual packets, APT achieves verification performance gains by orders of magnitude.

Solving preemptive job-shop scheduling problems using a true concurrency model

A true concurrency model called stopwatch durational action timed automata (S-DATA) is proposed to model preemptive timed concurrent systems. We demonstrate its usefulness in tackling the preemptive job-shop scheduling problem (PJSSP). This yields a compact reachability tree representing the possible schedules such that the durations and preemptions of the scheduled actions are dynamically managed. Different techniques are demonstrated to reduce the space search over such reachability tree, preserving the optimal schedules of the problem. The efficiency of approach is tested on a number of well-known benchmark problems and compared with the best method of scheduling in common use.

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

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

A Mobile-Agent Environment for Service Oriented System using Strong Mobility

In this research, we extended our existing work by applying the concept of strong mobility of mobile agent to make service oriented systems. Initially, an algorithm is developed on the model, based on strong mobility between the clients and server using mobile agent technology via VSAT (very small aperture technology). Where clients are deployed in critical regions which calculate flood discharge based on speed of flood. The proposed algorithm has been verified through the mobile Petri net, a formal language, which is used to formalize it. Reachability tree method of analysis is adopted to verify this model. For practical implementation, a critical area ‘Kalabagh’ is selected and a Java based system is developed of this model. The system is verified by using the historical data values of this critical area and the results verify the model.

A Method of Sample Models of Program Construction in Terms of Petri Nets

In the article a method of automated construction of Petri nets simulating the behaviour of imperative programs is considered from the formal point of view. Petri net samples with certain characteristics are necessary in programming new algorithms for program analysis; in particular, they can be used for developing or optimizing algorithms of Petri nets compositions and decompositions, building the reachability tree, checking invariants and so on. The generation process consists of two stages. At the first stage, construction templates for a resulting net and parameters for construction are described. With the help of these parameters it is possible to regulate the final size and the absolute or relative amount of certain structures in the resulting net. At the second stage, iterative process of automated net construction is used for Petri net generation of any size, limited only by an available computer memory. In the first section of the article the minimum necessary definitions are given and a new version of Petri nets composition operation by places is introduced. Commutative and associative properties of introduced binary operation allow to synchronize any number of Petri nets in arbitrary order. Then construction template is defined as a marked Petri net with input and output interfaces and rules for templates composition using this interfaces. A number of construction templates can be united in a collection, for which the evolution rules are defined. The completeness property of a collection guarantees that the collection evolution results in a Petri net that simulates the imperative program behavior. The article provides a version of the construction templates complete collection and an example of Petri net simulating sequential imperative program construction.

A reduced reachability tree for a class of unbounded petri nets

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

Lazy abstraction for timed automaton with interpolants

In this paper, we adapt lazy abstraction with interpolants for reachability analysis of timed automaton. It involves computing an abstract reachability tree whose nodes are labeled by a location and a given set of predicates. The set of predicates over‐approximates the reachable clock valuations of the corresponding location, and the approximation is refined when a spurious counterexample appears. A more precise approximation usually contains fewer spurious counterexamples that require refinement, but as a result it may delay the convergence of the algorithm. The state‐of‐the‐art lazy abstraction algorithms employ interpolants to produce an approximation that is closer to the actual reachable clock valuation to eliminate the spurious errors. However, interpolant also cannot ensure the quality of the resulting approximation, as it blindly relies on the SMT solver. In order to produce optimal approximation, we assign to each node a maximal lower and upper bounds (LU‐bounds) function to produce a node‐specific LU‐bounds abstraction; the abstraction can further over‐approximate the reachable clock valuations without introducing extra spurious counterexamples. Furthermore, in our algorithm, LU‐bounds are refined only when a spurious counterexample occurs. Hence, only those clock valuations that lead to error nodes are excluded. An alternative benefit of the LU‐bounds abstraction is that it ensures that the abstraction‐refinement loop terminates. © 2015 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Modelling safety of gantry crane operations using Petri nets

Being a powerful tool in modelling industrial and service operations, Petri net (PN) has been extremely used in different domains, but its application in safety study is limited. In this study, we model the gantry crane operations used for industrial activities using generalized stochastic PNs. The complete cycle of operations of the gantry crane is split into three parts namely inspection and loading, movement of load, and unloading of load. PN models are developed for all three parts and the whole system as well. The developed PN models have captured the safety issues through reachability tree. The hazardous states are identified and how they ultimately lead to some unwanted accidents is demonstrated. The possibility of falling of load and failure of hook, sling, attachment and hoist rope are identified. Possible suggestions based on the study are presented for redesign of the system. For example, mechanical stoppage of operations in case of loosely connected load, and warning system for use of wrong buttons is tested using modified models.

Time-Feasible Reachability Tree for Noncyclic Scheduling of Timed Petri Nets

Petri nets are useful for modeling and analyzing complex scheduling problems of automated manufacturing systems, such as robotized cluster tools for semiconductor manufacturing and robot cells with diverse and complex tool architectures and scheduling requirements. While there are many scheduling works on cyclic operation cycles of such systems and their Petri net models, automated manufacturing systems have significant noncyclic operation cycles. For instance, cluster tools cannot repeat identical work cycles for start-up and close-down operations of a lot, lot switching, nonidentical wafers, and too small lots. Such noncyclic scheduling problems can also be well modeled by timed Petri nets (TPNs) and can be solved by a branch and bound procedure that explores feasible states by branching and deleting the corresponding nodes in the reachability tree. The tree in general tends to be large, which significantly limits the computational efficiency. The tree generates nodes for all feasible markings regardless of the time evolution information associated with token holding times or firing delays. In this paper, we develop a way of significantly reducing the reachability tree by deleting the nodes or states that are infeasible in view of time evolution. To do this, we propose a time-feasible reachability tree that generates only time-feasible solutions under the earliest starting policy. We then use it for a branch and bound procedure for scheduling a TPN. We demonstrate its computational efficiency improvement with linear cluster tool scheduling problems.

Flexible Manufacturing System Design and Optimization using Petri Net-based Elementary Siphons

computer simulation system, which is based-siphon Petri net for control of concurrent processes of the reusable resources, that is mainly aimed to provide guidelines for control and management engineering of flexible manufacturing systems (FMSs) is presented in this work. We present an important technique for the analysis of siphon Petri net (PN) based on structural analysis, control, and simulation of the reachability tree used PN-tool with MATLAB. We are representing the deadlock prevention problem, which can be solved using the concept of Petri nets based siphons. Structural deadlock prevention is presenting for supervisors a class of FMS in simulating the system behaviour and elementary siphons control rule's inference. We are recovering the satiation of the liveness system from the consequence of analysis net by adding control place (monitor). Our experiment, a few monitors is added to the net for every minimal siphon taken from the consequence of the siphon in order to be liveness (i.e. deadlock-freeness), proposing extensions of the structural analysis of Petri nets, where deadlocks are regarding to the unmarked siphons. The siphon is recognized in the structural analysis of PN and can be used to control deadlocks in resource systems modelled.

A Survey of Reachability Trees of Unbounded Petri Nets

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

Heuristics on Reachability Trees for Bicriteria Scheduling of Stream Graphs on Heterogeneous Multiprocessor Architectures

In this article, we partition and schedule Synchronous Dataflow (SDF) graphs onto heterogeneous execution architectures in such a way as to minimize energy consumption and maximize throughput. Partitioning and scheduling SDF graphs onto homogeneous architectures is a well-known NP-hard problem. The heterogeneity of the execution architecture makes our problem exponentially challenging to solve. We model the problem as a weighted sum and solve it using novel state space exploration inspired from the theory of parallel automata. The resultant heuristic algorithm results in good scheduling when implemented in an existing stream framework.

Avoiding state explosion in a class of Petri nets

This paper introduces leveled Petri nets (PNs), and proposes a novel PN analysis tool, the superposition chain (SC), to avoid state explosion. It also introduces underlying tools—superposition and the leveled token game—to tackle the P vs NP problem, a well known problem in CS/AI community. The leveled token game, defined over a leveled PN, generates the SC of the PN. The leveling is based on the transitions such that a transition and all its input places are in the same level, and that there is no causality among transitions in a level, while transitions across levels indicate causality. The enabling rule is extended by superposition and firing history. Superposition of markings is defined by a set ⊻M of places p marked in superposition, and denotes that each p in ⊻M is marked individually, yet it is uncertain if all p in ⊻M are marked together. In other words, superposition loses which p in ⊻M is marked by conflicting transitions, which are revealed by the transition firing history. The firing history of p∈⊻M is also defined by a set, h(p), and denotes transition firings participated in p∈⊻M, yet does not enumerate their firing sequences to avoid the state explosion. Then, the compound firing history defined over ⊻M,h(⊻M), is used to reveal all conflicting transitions participated in ⊻M. Hence, ⊻M is not coverable as a whole, if there are conflicting firings in h(⊻M), which is used for the transition enabling. Consequently, the SC, generated by the leveled token game, specifies the PN behavior, as a reachability tree, generated by the (conventional) token game, also specifies a PN behavior.

Using Time Reachability Tree Logic to Specifying and Verifying Temporal Behavior of Workflow-Net

Workflow management has been a hot issue in both academic and industrial research. Deadline assignment is of great significance in workflow management. In order to avoid deadline violation, this paper presents an approach to the schedulability analysis of workflow system modeled in p-time Petri nets by separating timing properties from other behavior properties. The analysis of behavioral properties is conducted based on the reachability graph of the underlying p-Time Petri net, whereas timing constraints are checked in term of absolute and relative firing domains. Our technique is based on a concept called clock-stamped state class (CS-class) and temporal logic. With the reachability graph generated based on CS-class, we can directly compute the end-to-end time delay in workflow execution. We have identified a class of well-structured p-time Petri nets such that their reachability can be easy analyzed and temporal behavior can be easy analyzed by time reachability tree logical.