Properties Of Petri Nets Research Articles

BDD-based Bounded Model Checking for Temporal Properties of 1-Safe Petri Nets

In the paper we present a bounded model checking for 1-safe Petri nets and properties expressed in LTL and the universal fragment of CTL, based on binary decision diagrams. The presented experimental results show that we have obtained a technique which performs better in some of the considered cases, in comparison with the existing SAT-based implementation. The results are also compared with standard BDD-based symbolic method.

Linear time analysis of properties of conflict-free and general Petri nets

We introduce the notion of a T - path within Petri nets, and propose to adopt the model of directed hypergraphs in order to determine properties of nets; in particular, we study the relationships between T -paths and firable sequences of transitions. Let us consider a Petri net P = 〈 P , T , A , M 0 〉 and the set of places with a positive marking in M 0 , i.e., P 0 = { p ∣ M 0 ( p ) > 0 } . If we regard the net as a directed graph, the existence of a simple path from any place in P 0 to a transition t is, of course, a necessary condition for the potential firability of t . This is sufficient only if the net is a state machine, where | • t | = | t • | = 1 for all t ∈ T . In this paper we show that the existence of a T -path from any subset of P 0 to a transition t is a more restrictive condition and is, again, a necessary condition for the potential firability of t . But, in this case: (a) if P is a conflict-free Petri net, this is also a sufficient condition, (b) if P is a general Petri net, t is potentially firable by increasing the number of tokens in P 0 . For conflict-free nets ( CFPN) we consider the following problems: (a) determining the set of firable transitions, (b) determining the set of coverable places, (c) determining the set of live transitions, (d) deciding the boundedness of the net. For all these problems we provide algorithms requiring linear space and time, i.e., O ( | P | + | T | + | A | ) , for a net P = 〈 P , T , A , M 0 〉 . Previous results for this class of networks are given by Howell et al. (1987) [20], providing algorithms for solving problems in conflict-free nets in O ( | P | × | T | ) time and space. Given a Petri net and a marking M , the well-known coverability problem consists in finding a reachable marking M ′ such that M ′ ≥ M ; this problem is known to be EXPSPACE hard (Rackoff (1978) [33]). For general Petri nets we provide a partial answer to this problem. M is coverable by augmentation if it is coverable from an augmented marking M 0 ′ of the initial marking M 0 : M 0 ′ ≥ M 0 and, for all p ∈ P , M 0 ′ ( p ) = 0 if M 0 ( p ) = 0 . We solve this problem in linear time. The algorithms for computing T -paths are incremental: it is possible to modify the network (adding new places, transitions, arcs, tokens), and update the set of potentially firable transitions and coverable places without recomputing them from scratch. This feature is meaningful when used during the interactive design of a system.

Token Variation Vector and Analysis of Some Properties of Petri Nets

In this paper is presented a new definition for Petri nets: the Token Variation Vector. This definition is presented as simple strategy for evaluation of some properties of Petri nets, based on analysis of token number variation on places when enabled transitions fire. Some basic uses of this definition is presented as examples, showing how it can be used in union with others definitions to ensure coverage, deadlock analysis, and others properties.

IDD-based model validation of biochemical networks

This paper presents efficient techniques for the qualitative and quantitative analysis of biochemical networks, which are modeled by means of qualitative and stochastic Petri nets, respectively. The analysis includes standard Petri net properties as well as model checking of the Computation Tree Logic and the Continuous Stochastic Logic. Efficiency is achieved by using Interval decision diagrams to alleviate the well-known problem of state space explosion, and by applying operations exploiting the Petri structure and the principle of locality. All presented techniques are implemented in our tool IDD-MC which is available on our website.

A Petri net representation of Bayesian message flows: importance of Bayesian networks for biological applications

This article combines Bayes' theorem with flows of probabilities, flows of evidences (likelihoods), and fundamental concepts for learning Bayesian networks as biological models from data. There is a huge amount of biological applications of Bayesian networks. For example in the fields of protein modeling, pathway modeling, gene expression analysis, DNA sequence analysis, protein---protein interaction, or protein---DNA interaction. Usually, the Bayesian networks have to be learned (statistically constructed) from array data. Then they are considered as an executable and analyzable model of the data source. To improve that, this work introduces a Petri net representation for the propagation of probabilities and likelihoods in Bayesian networks. The reason for doing so is to exploit the structural and dynamic properties of Petri nets for increasing the transparency of propagation processes. Consequently the novel Petri nets are called "probability propagation nets". By means of examples it is shown that the understanding of the Bayesian propagation algorithm is improved. This is of particular importance for an exact visualization of biological systems by Bayesian networks.

A Petri net-based heuristic for simple assembly line balancing problem of type 2

In this paper, a Petri net-based heuristic is presented to solve simple assembly line balancing problem type-II (SALBP-2). Petri net is a mathematical and graphical tool to model and analyze for discrete event systems. SALBP-2 minimizes cycle time for given number of workstations. The presented heuristic determines available tasks and assign task to current workstation by using reachability analysis, one of the main properties of Petri nets, and token movement. Solution of SALBP-2 is implemented by iteratively solving the problem for several trial cycle time. If the cycle time is infeasible for given number of workstations, the heuristic increases the cycle time by a value until finding a feasible solution. To improve the solution, a binary search procedure is implemented between the first feasible solution and the last infeasible solution. Three versions of the heuristic are developed by integrating with forward, backward, and bidirectional procedures. All of them are coded in MATLAB, and their efficiencies are tested on benchmark datasets with 302 instances. Also a comparison study is made with a direct procedure and five heuristics based on differential evolution algorithm. Test and comparison results show that the proposed heuristic obtains good results for SALBP-2. Its performance is superior especially in large assembly lines.

Removing dead transitions in timed-arc Petri nets

Timed-arc Petri nets (TAPNs) are a timed extension of Petri nets where tokens are assigned an age indicating the time elapsed from its creation, and PT-arcs (place to transition arcs) are labelled with time intervals that are used to restrict the age of the tokens that can be used to fire the adjacent transition. This is a rather pathological model, as reachability is undecidable, whereas some other known properties of Petri nets, like boundedness, coverability and even termination, are decidable. This article focuses on the problem of detecting dead transitions, i.e. transitions that can be removed from the model since they can never become enabled. We prove that this problem is decidable for TAPNs with natural times, and we present an algorithm that can be used to find dead transitions in the particular case of 1-safe TAPNs.

Supervisory Controller Design to Enforce Some Basic Properties in Timed Petri Nets

Supervisory controller design to enforce boundedness, reversibility, and liveness in timed Petri nets with firing durations is considered. It is assumed that both controllable and uncontrollable transitions may be present and more than one transitions may fire simultaneously. The approach of stretching is used to represent the state of the system. Algorithms are presented to design a controller to enforce boundedness, reversibility, and liveness simultaneously.

A Novel Method for Reachability Determination in Petri Nets

Reachability is one of the most important behavioral properties of Petri nets and the past four decades have witnessed great efforts in developing various implementable methodologies in determining reachability of Petri nets. We propose in this paper a novel method for solving the fundamental equation in the reachability analysis, which has been known to be NP-complete. More specifically, by adopting a revised version of the cell enumeration method for an arrangement of hyperplanes in discrete geometry, we develop an efficient solution scheme to identify firing count vector solution(s) to the fundamental equation on a bounded integer set, with a complexity bound of O (( n ω) n–m ), where n is the number of transitions, m is the number of places and ω is the upper bound of the number of firings for every transition.

Modeling of the U1 snRNP assembly pathway in alternative splicing in human cells using Petri nets

The investigation of spliceosomal processes is currently a topic of intense research in molecular biology. In the molecular mechanism of alternative splicing, a multi-protein–RNA complex – the spliceosome – plays a crucial role. To understand the biological processes of alternative splicing, it is essential to comprehend the biogenesis of the spliceosome. In this paper, we propose the first abstract model of the regulatory assembly pathway of the human spliceosomal subunit U1. Using Petri nets, we describe its highly ordered assembly that takes place in a stepwise manner. Petri net theory represents a mathematical formalism to model and analyze systems with concurrent processes at different abstraction levels with the possibility to combine them into a uniform description language. There exist many approaches to determine static and dynamic properties of Petri nets, which can be applied to analyze biochemical systems. In addition, Petri net tools usually provide intuitively understandable graphical network representations, which facilitate the dialog between experimentalists and theoreticians. Our Petri net model covers binding, transport, signaling, and covalent modification processes. Through the computation of structural and behavioral Petri net properties and their interpretation in biological terms, we validate our model and use it to get a better understanding of the complex processes of the assembly pathway. We can explain the basic network behavior, using minimal T-invariants which represent special pathways through the network. We find linear as well as cyclic pathways. We determine the P-invariants that represent conserved moieties in a network. The simulation of the net demonstrates the importance of the stability of complexes during the maturation pathway. We can show that complexes that dissociate too fast, hinder the formation of the complete U1 snRNP.

Stochastic evaluation of supply chains and replenishment policies with Petri net components

The outbound logistics deals with the transportation and storage of finished goods. It is one of the most complex and expensive activity of the supply chain management and is decisive for the Quality of Service (QoS). This paper proposes the use of pre-defined Petri net components to model and evaluate this activity. The model that represents an outbound logistics scenario is obtained with a bottom-up approach, guaranteeing some expected Petri nets properties. Furthermore, the use of pre-defined, validated components allows modelling and evaluating the supply chain entities and processes through a systematic procedure. In the end of this paper, results related to a real case study, conducted by considering the proposed models, is depicted.

Deadlock-Freeness Analysis of Continuous Mono-T-Semiflow Petri Nets

Most verification techniques for highly populated discrete systems suffer from the state explosion problem. The "fluidification" of discrete systems is a classical relaxation technique that aims to avoid the state explosion problem. Continuous Petri nets are the result of fluidifying traditional discrete Petri nets. In continuous Petri nets the firing of a transition is not constrained to the naturals but to the non-negative reals. Unfortunately, some important properties, as liveness, may not be preserved when the discrete net model is fluidified. Therefore, a thorough study of the properties of continuous Petri nets is required. This paper focuses on the study of deadlock-freeness in the framework of mono-T-semiflow continuous Petri nets, i.e., conservative nets with a single repetitive sequence (T-semiflow). The study is developed both on untimed and timed systems. Topological necessary conditions are extracted for this property. Moreover, a bridge relating deadlock-freeness conditions for untimed and timed systems is established

COMPOSITIONAL VERIFICATION FOR WORKFLOW NETS

A workflow is the automation of business processes that describe activities in a business context. Petri nets have been widely used as a workflow modeling technique. Workflow nets (WF-nets), introduced by van der Aalst [LNCS1248 (1997) 407–426], are a class of Petri nets tailored towards workflow analysis. Soundness is used as the least correctness criterion for WF-nets in order to ensure that a process can terminate properly. This paper extends WF-nets by considering resources and allowing for multi-cases in a process. For the extended model, the relationship between soundness and standard behavior properties of Petri nets is investigated. A compositional method to construct and verify sound WF-nets is introduced.

System petri net specification

The high-level synthesis of digital systems, including controllers, DPSs, peripheral components, integrated microsystems, requires intermediate model representation of system specification. A few approaches have been proposed to use Petri nets as an intermediate system representation. The paper discusses essential Petri net properties and concentrates on requirements of the system representation. A new system Petri net text format is presented and it has been developed to support fully wide range of system specifications.

ANALYSIS OF SAFENESS, LIVENESS AND PERSISTENCE PROPERTIES OF PETRI NETS BY MEANS OF MONOTONE LOGIC FUNCTIONS

In the paper a new method of analysis of safeness, liveness and persistence properties of Petri nets is presented. This method is based on hierarchical representation of the state space of a Petri net by means of a collection of monotone logical functions and with related set of binary decision diagrams. The algorithm of symbolic analysis of a Petri net based behavioural specification of logic controllers is given, together with conditions, which allow analysing the properties mentioned above. One of the advantages of the proposed method is a possibility of detail analysis of selected local parts of a Petri net, as well as hierarchical approach for constructing the total state space.

Compositional analysis of Petri nets

Foundations of compositional analysis of Petri nets are presented. This analysis consist of the determination of properties of a given Petri net from the properties of its functional subnets. Compositional analysis covers the investigation of behavioral and structural properties of Petri nets with the help of matrix methods that use fundamental equations and invariants. The exponential acceleration of computations as a function of the dimensionality of a net is obtained.

Petri Net Refinement and Its Application in System Design

A scheme is obtained by using some kinds of Petri net refinement, according to the design of flexible manufacturing system. Two kinds of refinement are obtained. Dynamic properties have been investigated. The sufficient and necessary conditions of liveness preservation, boundedness preservation and reversibility preservation are presented. A flexible manufacturing system has been designed and verified. These results are useful for studying the static and dynamic properties of Petri nets and analyzing properties of large complex system. The refinement method is especially fit for the design of flexible manufacturing system and practical to use in reality.

On the Analysis of Some Structural Properties of Petri Nets

Some structural properties of a general Petri net (PN) are considered. The paper endeavors to improve the link between PNs, the theory of matrix analysis, and linear inequalities. Necessary and/or sufficient conditions for consistency, conservativeness, boundedness, and repetitiveness are given in terms of certain determinants of the incidence matrix of the net. Besides, when the incidence matrix is square, theorems are derived in terms of the modified-incidence-matrix eigenvalues and, thus, inherently reduce the state-space explosion problem. Examples are worked out to illustrate the results.

Fuzzy timed petri net definitions, properties, and applications

We present a new fuzzy timed Petri net model. Each transition firing in our model is associated with a fuzzy number; during transition, firings tokens are removed from input and added to output places. We consider the marks changing rate in each place as constant, and our performance analysis is based on the reachability state graph. Our model has fuzzy differential equations incorporated with it. Finally, we present the capabilities of the model by examples.

Structural and dynamic changes in concurrent systems: reconfigurable Petri nets

The aim of this work is the modeling and verification of concurrent systems subject to dynamic changes using extensions of Petri nets. We begin by introducing the notion of net rewriting system. In a net rewriting system, a system configuration is described as a Petri net and a change in configuration is described as a graph rewriting rule. We show that net rewriting systems are Turing powerful, that is, the basic decidable properties of Petri nets are lost and, thus, automatic verification in not possible for this class. A subclass of net rewriting systems are reconfigurable Petri nets. In a reconfigurable Petri net, a change in configuration amounts to the modification of the flow relations of the places in the domain of the involved rule according to this rule, independently of the context in which this rewriting applies. We show that reconfigurable Petri nets are formally equivalent to Petri nets. This equivalence ensures that all the fundamental properties of Petri nets are still decidable for reconfigurable Petri nets and this model is thus amenable to automatic verification tools. Therefore, the expressiveness of both models is the same, but, with reconfigurable Petri nets, we can easily and directly model systems that change their structure dynamically.