Analysis of Petri Nets and Transition Systems

When a Petri net is synthesised from a labelled transition system, it is frequently desirable that certain additional constraints are fulfilled. For example, in circuit design, one is often interested in constructing safe Petri nets. Targeting such subclasses of Petri nets is not necessarily computationally more efficient than targeting the whole class. For example, targeting safe nets is known to be NP-complete while targeting the full class of place/transition nets is polynomial, in the size of the transition system. In this paper, several classes of Petri nets are examined, and their suitability for being targeted through efficient synthesis from labelled transition systems is studied and assessed. The focus is on choice-free Petri nets and some of their subclasses. It is described how they can be synthesised efficiently from persistent transition systems, summarising and streamlining in tutorial style some of the authors’ and their groups’ work over the past few years.

Knowledge-based approach to Petri nets analysis

In this chapter, modeling of GRNs using Petri net theory is considered. It aims at providing a conceptual understanding of Petri nets to enable the reader to explore GRNs applying Petri net modeling and analysis techniques. Starting with an overview on modeling biochemical networks using Petri nets, the state-of-the-art with focus on GRNs is described. Other modeling techniques, for example, hybrid Petri nets are discussed. Basic concepts of Petri net theory are introduced involving special analysis techniques for modeling biochemical systems, for example, MCT-sets, T-clusters, and Mauritius maps. To illustrate these Petri net concepts, a more complex case study–the gene regulation in Duchenne Muscular Dystrophy–is explained in detail, considering the biological background and the interpretation of analysis results. Considering both, advantages and disadvantages, the chapter demonstrates the usefulness of Petri net modeling, in particular for GRNs.

Pre-synthesis of Petri nets based on prime cycles and distance paths

CORBA is a standard proposed by the Object Management Group (OMG) that promotes interoperability between distributed object systems. Following the standardization of this object-oriented middleware, the OMG has specified a set of Common Object Services (COS) that are meant to serve as the building blocks of distributed CORBA applications. The COSes are specified using CORBA Interface Definition Language (IDL), that describes the syntactic aspects of services supported by remote objects. However, CORBA-IDL does not support specification of the behaviour of objects in an abstract and formal way, and behavioural specification is mostly provided in plain English. To overcome this problem, we have proposed a formal description technique (Cooperative Objects) based on high-level Petri nets, and developed a software support environment. The goal of this paper is to demonstrate that our approach is suited to the formal specification of typical CORBA COS, and presents a Cooperative Object model of the CORBA event service, a COS that provides asynchronous, one-to-many communication between objects. The advantages of dealing with a tool-supported, executable formal notation are detailed, as well as the results that can be obtained through Petri net analysis techniques.Key wordsDistributed systemsbehavioural specificationCORBAhigh-level Petri nets

Problems such as low transport efficiency, low safety factor, and difficulty in the dispatching of hybrid locomotives, which are not optimal in the automated transport of coal mines, are always difficult to study. These serious situations are studied and avoided through petri net (PN) analysis techniques in this paper. Firstly, by analyzing the relationship between locomotives, road sections and elements of petri net, petri net scheduling mode is built for locomotives with three different routes according to three different mine locomotive scheduling methods. Then, based on a certain set of resources in PN, the concepts of a deadlock state equation and a potential deadlock state equation are introduced. The deadlock state equation describes the intrinsic relationship between resources assignation and a deadlock state in PN. This equation is a necessary and sufficient condition for the occurrence of a deadlock situation. In the following part, the deadlock equations of several network models are solved, and the deadlock verification is carried out using the software PIPE, which proves the validity of the equations. The results show that the solution of the deadlock equation and the potential deadlock equation can effectively avoid the deadlock of petri net model, so as to prevent the blocking and collision of locomotives. At the same time, the whole modeling process also verifies the effectiveness of petri net for mining bottom locomotive modeling.

Timed Petri nets can be used to model and analyse scheduling problems. To support the modelling of scheduling problems, we provide a method to map tasks, resources and constraints onto a timed Petri net. By mapping scheduling problems onto Petri nets, we are able to use standard Petri net theory. In this paper we will show that we can use Petri net based tools and techniques to find conflicting and redundant precedences, upper- and lower-bounds for the makespan, etc. This is illustrated by a Petri net based analysis of the notorious 10×10 problem due to Fisher & Thompson (1963)

This paper proposes a fully expanded tree (FET) approach for one-place-unbounded Petri nets. The FET of a one-place-unbounded Petri net consists of all and only reachable markings from its initial marking. Its applications to liveness and deadlock analysis for such Petri nets are developed. The proposed method has a larger application scope than all the existing methods for them. Several examples are provided to show its superiority over the state-of-the-art methods.

The paper presents an extensible Petri Net Markup Language (xPNML), which is an extended version of PNML. The xPNML format overcomes limitations associated with PNML structure for simulation and analysis of stochastic Petri nets. In addition, the proposed format supports modular concepts for creating specific models (for example, a module for machine operation including working, failed and repair states). These modules can be used as templates and exchanged among different net models. The paper provides the ontology of xPNML model and overview of its JAVA based implementation for simulation and analysis of stochastic Petri nets.

In this paper, we investigate transition systems of a class of Petri nets suitable for the modelling and behavioural analysis of globally asynchronous locally synchronous systems. The considered model of Elementary Net Systems with Localities (ENL-systems) is basically that of Elementary Net Systems (EN-systems) equipped with an explicit notion of locality. Each locality identifies a distinct set of events which may only be executed synchronously, i.e., in a maximally concurrent manner. For this reason, the overall behaviour of an ENL-system cannot be represented by an interleaved transition system, with arcs being labelled by single events, but rather by a suitable notion of a step transition system, with arcs being labelled by sets of events executed concurrently.We completely characterise transition systems which can be generated by Elementary Net Systems with Localities under their intended concurrency semantics. In developing a suitable characterisation, we follow the standard approach in which key relationships between a Petri net and its transition system are established via the regions of the latter defined as specific sets of states of the transition system. We argue that this definition is insufficient for the class of transition systems of ENL-systems, and then augment the standard notion of a region with some additional information, leading to the notion of a region with explicit input and output events (or io-region).We define, and show consistency of, two behaviour preserving translations between ENL-systems and their transition systems. As a result, we provide a solution to the synthesis problem of Elementary Net Systems with Localities, which consists in constructing an ENL-system for a given transition system in such a way that the transition system of the former is isomorphic to the latter.KeywordsTheory of concurrencyPetri netselementary net systemslocalitiesanalysis and synthesisstep sequence semanticsstructure and behaviour of netstheory of regionstransition systems

Petri nets play a central role in the analysis of discrete, parallel and distributed systems. Reachability problem is one of the analysis of Petri nets. This paper address the analysis of reachable marking of petri nets using invariants. We translate Petri nets as transition systems. Then, through computing vanishing ideal of sample points, we present a new approach for constructing polynomial equation invariants of Petri nets. Our approach avoids first-order quantifier elimination and cylindrical algebraic decomposition. From the preliminary experiment results, we demonstrate the feasibility of our approach.

The concept of siphons plays an important role in the analysis of Petri nets. In particular, criteria for liveness and reachability of some subclasses of Petri nets can be stated in terms of siphons. However, the computation of these structural components can be time consuming or even impossible. Recently, a variety of deadlock control policies that rely on partial siphon computation and enumeration have been developed. We propose a polynomial algorithm to find a set of elementary siphons in a class of Petri nets called systems of simple sequential processes with resources, S/sup 3/PR for short. Based on them, other siphons, called dependent ones, can be composed by simple set operations. Case studies show that the proposed algorithm is more computationally efficient than INA, integrated net analyzer, a widely used Petri net analysis tool.

This paper introduces a new matrix called the sign incidence matrix for Petri nets. Using this sign incidence matrix, we present a simple algorithm for generating all basis siphons or traps without first generating all siphons or traps. Any siphon (trap) can be expressed as an union of basis siphons (basis traps). The concept of siphons and traps plays an important role in the analysis of Petri nets. In particular, criteria for liveness and reachability of some subclasses of Petri nets can be stated in terms of siphons and traps.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Given an initial marking and final marking for a Petri net model, an optimal firing sequence problem is defined as the problem to find an optimal transition firing sequence to minimize the objective function. For the purpose of analysis of general integer programming problems, we propose a Petri net representation and reachability analysis of integer programming problems. In the proposed method, an integer programming problem is converted into the optimal firing sequence problem of Petri nets. By utilizing the proposed algorithm, integer programming problems can be visualized and analyzed by the Petri net theory. We apply the proposed methodology to the scheduling problems of dual armed cluster tools. The valid inequalities are derived from the reachability analysis. Numerical results show that those valid inequalities can significantly reduce the computational time of the original integer programming problem.

Based on the workflow management coalition (WFMC) standard and component software technologies, this paper addresses our Petri-nets-based approach for workflow and process automation. Petri nets provide graphical and mathematical formalisms for work-flow process definition and analysis. In this paper, we will present how to use Petri nets and a toolkit NCUPN (National Central University Petri Nets toolkit) for process definition and analysis. NCUPN is a Petri nets modeling and analysis toolkit developed to help software engineers in drawing and doing analysis. Once a workflow process is drawn using NCUPN, analysis can be done automatically. In this paper, we will demonstrate how to use NCUPN to analyze behavioral properties of workflow process, such as deadlock, liveness, safety, and starvation.