Structural Analysis of Petri Nets

<u>1.1 The language CAP</u>While there are a couple of programming languages which allow the representation of parallelism /1,2,5/ there is a lack for languages which allow in an as well disciplined as general manner the description of concurrency.A well defined mathematical description method for concurrent processes is given by the Petri net model /3,7,9/. It fulfils the above requirements and in addition is widely used and extremely easy to understand.This concept is integrated into the language CAP in a very elegant way: Lables are used as places, where one has in mind that a labelled statement puts a token into its label(s) <u>after</u> it has been processed. On the other hand the processability of a statement is controlld by "On-conditions" on labels. Every statement is processed if and only if its "On-condition" has become true. At this time, in accordance with the firing-rule (="On-condition"), "tokens" are withdrawn from certain labels used within the "On-condition" and placed into the labels of the statement.In addition various parameters may be associated to transitions thus offering a description power similar to the Macro-E-Nets /8/. On the other hand there are a lot of language-constructs to support structured programming.<u>1.2 The analysis of CAP-programs at compile-time</u>Since a couple of years Petri nets are object of research. For some rather restricted subclasses of Petri nets sufficiant an necessary conditions for the topology of a net to be well-formed are known /3,4,7/. Under a well-formed net we understand a net <u>without dead-locks</u> being <u>safe</u> (no loss of information) and <u>residue-free</u> (independent from a certain history). Related to results by Herzog an Yoeli /4/ for a subclass of Petri nets we prove that there are six conditions for the topology of a net of the subclass of Petrinets which are used to describe the control-structure of CAP-programs so that the net is well-formed.By this we can decide at compile-time without simulation whether a CAP-program describes a life, safe and residue-free system (i. e. a useful one) or not. After having defined, what we mean ba a "Structured CAP-program" we will have the fine result that every structured CAP-program is well-formed.<u>1.3 The simulation of systems described in CAP</u>As CAP allows the description of systems down to a level of specification which is comparable with the bit-level of digital systems inclusive the detailled description of the real-time-behaviour, for the processing of CAP one needs a system similar to a simulator for digital circuits. On the other hand also very global descriptions on a high level of abstraction are possible.This implies that the run-time system for CAP must be extremely adaptive. We tried to solve this problem by a very flexible and powerful table driven and event oriented simulator with the event-mechanism being directly adopted from the Petri net concept of CAP.

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.

Starts with a brief review of the history and the application areas considered in the literature. The author then proceeds with introductory modeling examples, behavioral and structural properties, three methods of analysis, subclasses of Petri nets and their analysis. In particular, one section is devoted to marked graphs, the concurrent system model most amenable to analysis. Introductory discussions on stochastic nets with their application to performance modeling, and on high-level nets with their application to logic programming, are provided. Also included are recent results on reachability criteria. Suggestions are provided for further reading on many subject areas of Petri nets.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This paper presents a heuristic search approach for scheduling flexible manufacturing systems (FMS) with due dates based on the Petri net state equation. It is known that the state equation of a Petri net has a solution if the goal marking can be reached from the initial marking. In terms of scheduling, this denotes that the costs of jobs can be obtained from the solution. In this paper, the idea of jobwise Petri nets is introduced. A jobwise net is a part of the model of an FMS. It is only associated with one job and the resources that the job uses. A model of N jobs can be split into N jobwise net models. We combine jobwise Petri nets and the state equations to implement the search algorithm. The objective is to minimize weighted tardiness (lateness), which is the difference of the completion time of a job and its due date. In order to show that this scheduling method can get near-optimal solutions, we compare it with some commonly used dispatching rules by simulations.

This study deals with deadlock problems in a system with uncontrollable and unobservable events, which is modeled with a subclass of Petri nets. We design monitors to enforce constraints for minimal siphons in a Petri net model. A key point of this paper is how to select the input and output arcs of a monitor when there exist uncontrollable and unobservable transitions. After the enforcement of constraints, an iterative algorithm is developed to obtain a controlled system with liveness. A sufficient and necessary condition is established to decide the existence of a monitor enforcing a constraint in a plant with uncontrollable and unobservable transitions.

A design methodology for self-timed VLSI systems is proposed. This includes a language for describing algorithmic behaviour which is based on the data-flow concept. A subclass of Petri nets is introduced to model the designed system, and a refinement technique is presented that allows for the synthesis of correct-by-construction systems. Finally, a systematic means for implementing the design is given.

Petri nets have traditionally been motivated by their ability to express concurrency. Subclasses of Petri nets with concurrency but without choices or conflicts have been studied extensively. One of the best known - and comparatively restricted - such class are the marked graphs [4]. However, perhaps the largest class of (intuitively) choice-free nets are the persistent nets [7], a class of nets that is significantly larger than marked graphs. Some early results about persistent nets are Keller's theorem [6], which will be recalled in a later part of this paper, and the famous semilinearity result of Landweber and Robertson [7], which states that the set of reachable markings of a persistent net is semilinear. In this paper, we show that in bounded persistent nets, the smallest cycles of its reachability graph enjoy a uniqueness property.

This chapter introduces six subclasses of Petri nets that are of beautiful structures: state machines, marked graphs, free-choice nets, asymmetric-choice nets, weakly persistent nets and normal nets. On the basis of their structural characteristics, some perfect conditions can be provided to decide some important properties such as their liveness and reachability.

In spite of their increasing popularity for modeling and performance analysis of parallel systems, Petri nets play only a marginal role in the area of synthesis of asynchronous hardware. It is suggested that a change in the perception of their role will lead to a cost effective design method for parallel asynchronous architectures. A hardware implementable subclass of Petri nets is presented that exploits conflicts as a nondeterministic scheduling feature and that offers token evaluation as an option for path selection. The net primitives, complemented with functional primitives, constitute the lowest abstractions for a data flow oriented design paradigm.

It is more difficult to model a discrete system than a continuous system. Recently, a great number of applications of Petri nets to the design and the analysis of discrete systems has been reported. However, a common problem in its applications is that the required computer memories and the computation times increase explosively in accordance with the increase in the number of systems components. Some methods to divide or to reduce Petri nets have been proposed to solve this problem. Although the liveness and boundedness of Petri nets are held in the divided or reduced Petri nets, the reachability problem cannot be solved by these methods.This paper proposes another method to model discrete systems by Petri nets with place invariants. A sequential control system will be described as a typical kind of discrete system, and its structural characteristics will be used in modeling. Each component of a sequential control system will be modeled by a sub‐Petri net with place invariants. There are many components in one sequential control system, but not each one is necessarily complicated. Most sub‐Petri nets do not have so many places or transitions. It is also well known that a Petri net with place invariants is bounded and can be live by placing sufficient tokens into its initial marking. Besides, the reachability problem is not so difficult to solve in a small sub‐Petri net. Further, an activating relation to combine two or more sub‐Petri nets is defined, and some rules to reduce conflicts among enabled transitions will be described for simulations.

Place/transition Petri nets are a standard model for a class of distributed systems whose reachability spaces might be infinite. One of well-studied topics is verification of safety and liveness properties in this model; despite an extensive research effort, some basic problems remain open, which is exemplified by the complexity status of the reachability problem that is still not fully clarified. The liveness problems are known to be closely related to the reachability problem, and various structural properties of nets that are related to liveness have been studied. Somewhat surprisingly, the decidability status of the problem of determining whether a net is structurally live, i.e. whether there is an initial marking for which it is live, remained open for some time; e.g. Best and Esparza (Inf Process Lett 116(6):423–427, 2016. https://doi.org/10.1016/j.ipl.2016.01.011 ) emphasize this open question. Here we show that the structural liveness problem for Petri nets is ExpSpace-hard and decidable. In particular, given a net N and a semilinear set S, it is decidable whether there is an initial marking of N for which the reachability set is included in S; this is based on results by Leroux (28th annual ACM/IEEE symposium on logic in computer science, LICS 2013, New Orleans, LA, USA, June 25–28, 2013, IEEE Computer Society, pp 23–32, 2013. https://doi.org/10.1109/LICS.2013.7 ).

Given a labeled Petri net with silent (unobservable) transitions, we are interested in performing initial marking estimation in a probabilistic setting. We assume a known finite set of initial markings, each with some a priori probability, and our goal is to obtain the conditional probabilities of initial markings of the Petri net, conditioned on an observed sequence of labels. Under a Markovian assumption on the probabilistic model, we develop a recursive algorithm that allows us to efficiently determine the conditional probabilities for each possible initial marking (conditioned on the sequence of observations seen so far). We illustrate the proposed methodology via an example and discuss potential applications in the context of initial state opacity for security applications.

This article presents a new way of generating a generalized state equation that is useful for analyzing the token flow of the Petri Net (PN) with priority. The transition values in the firing vector as used in the conventional state equation are replaced with transition variables, which are generated by multiplying a series of firing condition functions taking the weighted inhibitor arc into account. The actual value of a transition variable is determined by taking priority and the present marking into account. The proposed state equation generalizes the conventional one by using the transition variable form and by containing the formulation of priority. Given the initial marking, the subsequent marking evolution can be determined successively from the generalized state equation as the simultaneous firings occur. A PN with deadlock is analyzed as an example to establish the validity of the generalized state equation. © 2003 Wiley Periodicals, Inc.

In this paper, we introduce a framework composed of a syntax and its compositional Petri net semantics, for the specification and verification of properties (like authentication) of security protocols. The protocol agents (e.g., an initiator, a responder, a server, a trusted third party, ...) are formalized as roles, each of them having a predefined behavior depending on their global and also local knowledge (including for instance public, private and shared keys), and may interact in a potentially hostile environment. The main characteristics of our framework, is that it makes explicit, structured and formal, the usually implicit information necessary to analyse the protocol, for instance the public and private contextof execution. The roles and the environment are expressed using SPL processes and compositionally translated into high-level Petri nets, while the context specifying the global and local knowledge of the participants in the protocol is used to generate the corresponding initial marking (with respect to the studied property). Finally, this representation is used to analyse the protocol properties, applying techniques of simulation and model-checking on Petri nets. The complete approach is illustrated on the case study of the Kao-Chow authentication protocol.

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.