On the Regularity of Petri Net Languages

Petri nets are known to be useful for modeling concurrent systems. Once modeled by a Petri net, the behavior of a concurrent system can be characterized by the set of all executable transition sequences, which in turn can be viewed as a language over an alphabet of symbols corresponding to the transitions of the underlying Petri net. In this paper, we study the language issue of Petri nets from a computational complexity viewpoint. We analyze the complexity of theregularity problem(i.e., the problem of determining whether a given Petri net defines an irregular language or not) for a variety of classes of Petri nets, includingconflict-free,trap-circuit,normal,sinkless,extended trap-circuit,BPP, andgeneralPetri nets. (Extended trap-circuit Petri nets are trap-circuit Petri nets augmented with a specific type ofcircuits.) As it turns out, the complexities for these Petri net classes range from NL (nondeterministic logspace), PTIME (polynomial time), and NP (nondeterministic polynomial time), to EXPSPACE (exponential space). In the process of deriving the complexity results, we develop adecomposition approachwhich, we feel, is interesting in its own right, and might have other applications to the analysis of Petri nets as well. As a by-product, an NP upper bound of the reachability problem for the class of extended trap-circuit Petri nets (which properly contains that of trap-circuit (and hence, conflict-free) and BPP-nets, and is incomparable with that of normal and sinkless Petri nets) is derived.     

Verification of bounded Petri nets using integer programming

Model checking based on the causal partial order semantics of Petri nets is an approach widely applied to cope with the state space explosion problem. One of the ways to exploit such a semantics is to consider (finite prefixes of) net unfoldings—themselves a class of acyclic Petri nets—which contain enough information, albeit implicit, to reason about the reachable markings of the original Petri nets. In [19], a verification technique for net unfoldings was proposed, in which deadlock detection was reduced to a mixed integer linear programming problem. In this paper, we present a further development of this approach. The essence of the proposed modifications is to transfer the information about causality and conflicts between the events involved in an unfolding, into a relationship between the corresponding integer variables in the system of linear constraints. Moreover, we present some problem-specific optimisation rules, reducing the search space. To solve other verification problems, such as mutual exclusion or marking reachability and coverability, we adopt Contejean and Devie's algorithm for solving systems of linear constraints over the natural numbers domain and refine it, by taking advantage of the specific properties of systems of linear constraints to be solved. Another contribution of this paper is a method of re-formulating some problems specified in terms of Petri nets as problems defined for their unfoldings. Using this method, we obtain a memory efficient translation of a deadlock detection problem for a safe Petri net into an LP problem. We also propose an on-the-fly deadlock detection method. Experimental results demonstrate that the resulting algorithms can achieve significant speedups.

基本并行进程活性的限界模型检测

基本并行进程是一个用于描述和分析并发程序的模型,是Petri网的一个重要子类.EG逻辑是一种在Hennessy-Milner Logic的基础上增加EG算子的分支时间逻辑,其中的AF算子表示从当前的状态出发性质最终会被满足,因此EG逻辑是能够表达活性的逻辑.然而,基于基本并行进程的EG逻辑的模型检测问题是不可判定的.由此,本文提出了基本并行进程上EG逻辑的限界模型检测方法.首先,给出了基本并行进程上EG逻辑的限界语义,然后采用基于约束的方法,将基本并行进程上EG逻辑的限界模型检测问题转化为线性整数算术公式的可满足性问题,最后利用SMT求解器进行求解.     

A deadlock prevention approach for flexible manufacturing systems without complete siphon enumeration of their Petri net models

Siphons are very important in the analysis and control of deadlocks in a Petri net. However, it is quite time-consuming or even impossible to get the complete siphon enumeration of a Petri net. This paper focuses on the deadlock prevention problems in flexible manufacturing systems that are modeled with S⁴PR, a general class of Petri nets. The analysis of S⁴PR leads us to characterize deadlock situations in terms of insufficiently marked siphons. The method proposed in this paper is an iterative approach. At each iteration, a non-max-marked siphon is computed by solving a mixed integer linear programming problem. Then the siphon is max-marked through a P-invariant by adding a monitor place. This process is carried out until no non-max-marked siphon can be found in the net. As a result all the siphons in the net are max-controlled. Then the net becomes live. Without computing all the siphons, a monitor-based liveness-enforcing Petri net supervisor can be found with more permissive behavior. A number of flexible manufacturing examples are used to demonstrate the proposed methods.     

Priority conflict-free Petri nets

A number of problems concerning priority conflict-free Petri nets are investigated in this paper. We show the reachability problem for such Petri nets to be NP-complete. (Using a similar technique, the NP-completeness result applies to the class of priority BPP-nets as well.) As for the boundedness problem, an NP-completeness result is demonstrated for priority conflict-free Petri nets with two types of prioritized transitions. (In contrast, the problem is known to be P-complete for conflict-free Petri nets without priorities.) We also investigate the home state problem, i.e., the problem of determining whether home states exist in a given a Petri net, for conflict-free Petri nets with and without priorities. As it turns out, home states always exist for bounded conflict-free Petri nets without priorities. If an additional liveness constraint is imposed, such Petri nets are guaranteed to be ‘reversible’ (i.e., their initial states are home states). For priority conflict-free Petri nets, being bounded and live is sufficient for the existence of home states. However, if the liveness assumption is dropped, the existence of home states is no longer guaranteed. Received: 1 April 1997 / 8 December 1997     

Identification of Petri Nets from Knowledge of Their Language

In this paper we deal with the problem of identifying a Petri net system, given a finite language generated by it. First we consider the problem of identifying a free labeled Petri net system, i.e., all transition labels are distinct. The set of transitions and the number of places is assumed to be known, while the net structure and the initial marking are computed solving an integer programming problem. Then we extend this approach in several ways introducing additional information about the model (structural constraints, conservative components, stationary sequences) or about its initial marking. We also treat the problem of synthesizing a bounded net system starting from an automaton that generates its language. Finally, we show how the approach can also be generalized to the case of labeled Petri nets, where two or more transitions may share the same label. In particular, in this case we impose that the resulting net system is deterministic. In both cases the identification problem can still be solved via an integer programming problem.

Decidability of model checking for infinite-state concurrent systems

We study the decidability of the model checking problem for linear and branching time logics, and two models of concurrent computation, namely Petri nets and Basic Parallel Processes. Received: 29 November 1994 / 15 November 1995     

Net structure and control logic synthesis of controlled Petri nets

Control logic synthesis of discrete-event systems is considered in the setting of controlled Petri nets. The problem is to find a control policy that restricts the behavior of a controlled Petri net so that a collection of forbidden state conditions is satisfied. S-decreases are introduced as a tool for the control synthesis. The S-decreases are weight vectors defined on the places of a net such that the weighted sum of tokens in the net never increases with any transition firing. On the basis of S-decreases, the authors propose an efficient method for the synthesis of the maximally permissive state feedback control polity for a class of controlled Petri nets whose uncontrolled subnets are forward and backward conflict-free nets. This method upgrades all integer linear programming-based methods for which one only requires to solve the much simpler linear programming problems to determine maximally permissive controls     

Steady-state performance evaluation of continuous mono-T-semiflow Petri nets

The number of states in discrete event systems can increase exponentially with respect to the size of the system. A way to face this state explosion problem consists of relaxing the system model, for example by converting it to a continuous one. In the scope of Petri nets, the firing of a transition in a continuous Petri net system is done in a real amount. Hence, the marking (state) of the net system becomes a vector of non-negative real numbers. The main contribution of the paper lies in the computation of throughput bounds for continuous Petri net systems with a single T-semiflow. For that purpose, a branch and bound algorithm is designed. Moreover, it can be relaxed and converted into a linear programming problem. Some conditions, under which the system always reaches the computed bounds, are extracted. The results related to the computation of the bounds can be directly applied to a larger class of nets called mono T-semiflow reducible.     

基于GMEC转换算法的Petri网结构控制器综合方法

针对含不可控变迁Petri网系统禁止状态控制器设计问题,提出了一种基于矩阵变换和整数线性规划的结构控制器综合方法。该方法的关键是对代表系统合法状态的广义互斥约束(generalized mutual exclusion constraint, GMEC)进行转换。首先,根据Petri网系统的关联矩阵,将库所集分为无关库所集、不可控库所集和补足库所集。其次,通过对非允许GMEC中补足库所的权值和不可控库所的权值进行处理,并运用整数线性规划将非允许GMEC转换为允许GMEC。在允许GMEC的基础上,根据库所不变量原理设计出Petri网系统的结构控制器。最后,以某零件加工系统为例验证了所提方法的泛用性和高效性,为实际智能制造系统的监督控制器设计提供有效参考方案。     

A Survey of Petri Net Methods for Controlled Discrete Event Systems

This paper surveys recent research on the application of Petri net models to the analysis and synthesis of controllers for discrete event systems. Petri nets have been used extensively in applications such as automated manufacturing, and there exists a large body of tools for qualitative and quantitative analysis of Petri nets. The goal of Petri net research in discrete event systems is to exploit the structural properties of Petri net models in computationally efficient algorithms for computing controls. We present an overview of the various models and problems formulated in the literature focusing on two particular models, the controlled Petri nets and the labeled nets. We describe two basic approaches for controller synthesis, based on state feedback and event feedback. We also discuss two efficient techniques for the on-line computation of the control law, namely the linear integer programming approach which takes advantage of the linear structure of the Petri net state transition equation, and path-based algorithms which take advantage of the graphical structure of Petri net models. Extensions to timed models are briefly described. The paper concludes with a discussion of directions for future research.     

Liveness-enforcing supervisors synthesis for a class of generalised Petri nets based on two-stage deadlock control and mathematical programming

In this article we deal with deadlock prevention problems for S⁴PR, a class of generalised Petri nets, which can well model a large class of flexible manufacturing systems where deadlocks are caused by insufficiently marked siphons. We present a deadlock prevention methodology that is an iterative approach consisting of two stages. The first one is called siphon control, which is to add for each insufficiently marked minimal siphon a control place to the original net. Its objective is to prevent a minimal siphon from being insufficiently marked. The second one, called control-induced siphon control, is to add a control place to the augmented net with its output arcs connecting to the source transitions, which assures that there are no new insufficiently marked siphons generated. At each iteration, a mixed integer programming approach is adopted for generalised Petri nets to obtain an insufficiently marked minimal siphon from the maximal deadly siphon. This way complete siphon enumeration is avoided that is much more time-consuming for a sizeable plant model than the proposed method. The relation of the proposed method and the liveness and reversibility of the controlled net is obtained. Examples are presented to demonstrate the presented method.     

SAT-Solving the Coverability Problem for Petri Nets

Net unfoldings have attracted great attention as a powerful technique for combating state space explosion in model checking, and have been applied to verification of finite state systems including 1-safe (finite) Petri nets and synchronous products of finite transition systems. Given that net unfoldings represent the state space in a distributed, implicit manner the verification algorithm is necessarily a two step process: generation of the unfolding and reasoning about it. In his seminal work McMillan (K.L. McMillan, Symbolic Model Checking. Kluwer Academic Publishers, 1993) showed that deadlock detection on unfoldings of 1-safe Petri nets is NP-complete. Since the deadlock problem on Petri nets is PSPACE-hard it is generally accepted that the two step process will yield savings (in time and space) provided the unfoldings are small.In this paper we show how unfoldings can be extended to the context of infinite-state systems. More precisely, we show how unfoldings can be constructed to represent sets of backward reachable states of unbounded Petri nets in a symbolic fashion. Furthermore, based on unfoldings, we show how to solve the coverability problem for unbounded Petri nets using a SAT-solver. Our experiments show that the use of unfoldings, in spite of the two-step process for solving coverability, has better time and space characteristics compared to a traditional reachability based implementation that considers all interleavings for solving the coverability problem.     

On decidability of LTL model checking for process rewrite systems

We establish a decidability boundary of the model checking problem for infinite-state systems defined by Process Rewrite Systems (PRS) or weakly extended Process Rewrite Systems (wPRS), and properties described by basic fragments of action-based Linear Temporal Logic (LTL) with both future and past operators. It is known that the problem for general LTL properties is decidable for Petri nets and for pushdown processes, while it is undecidable for PA processes.We show that the problem is decidable for wPRS if we consider properties defined by LTL formulae with only modalities strict eventually, strict always, and their past counterparts. Moreover, we show that the problem remains undecidable for PA processes even with respect to the LTL fragment with the only modality until or the fragment with modalities next and infinitely often.     

An Optimization Approach to Petri Net Monitor Design

This note addresses the problem of enforcing generalized mutual exclusion constraints on a Petri net plant. First, we replace the classical partition of the event set into controllable and uncontrollable events from supervisory control theory, by associating a control and observation cost to each event. This leads naturally to formulate the supervisory control problem as an optimal control problem. Monitor places which enforce the constraint are devised as a solution of an integer linear programming problem whose objective function is expressed in terms of the introduced costs. Second, we consider timed models for which the monitor choice may lead to performance optimization. If the plant net belongs to the class of mono-T-semiflow nets, we present an integer linear fractional programming approach to synthesize the optimal monitor so as to minimize the cycle time lower bound of the closed loop net. For strongly connected marked graphs the cycle time of the closed-loop net can be minimized     

Investigation of Maximum and Minimum Satisfiability Problems Using L-Partition

Problems of the maximum and the minimum satisfiability on the basis of the integer linear programming and L-partition are studied. The L-structure of polyhedrons of the problems is investigated. Families are set up of unweighted problems of the maximum and the minimum satisfiability, the powers of L-coverings of which grow exponentially with an increase in the number of variables in a formula.     

Petri nets and regular languages

It is shown that the regularity problem for firing sequence sets of Petri nets is decidable. For the proof, new techniques to characterize unbounded places are introduced. In the class $\text{L}$₀ of terminal languages of labelled Petri nets the regularity problem in undecidable. In addition some lower bounds for the undecidability of the equality problems in $\text{L}$₀ and $\text{L}$ are given. L₀^λ is shown to be not closed under complementation without reference to the reachability problem.     

用于一致性检查的知识库Petri网模型

知识库是一致性是决定专家系统效率及求解正确性的关键因素。本文以Ｐｅｔｒｉ网为工具对知识库进行模拟分析，把知识库一致性的检查化简为线性代数问题，把这一方法应用于分布式知识库系统，首次得到了检查其一致性的形式方法。本文最后给出了一致性检查的充分必要条件，为建立（分布式）知识库的自动维护系统打下了基础。     

The synthesis problem of Petri nets

The synthesis problem of concurrent systems is the problem of synthesizing a concurrent system model from sequential observations. The paper studies the synthesis problem for elementary Petri nets and transition systems. A characterization of the class of transition systems which correspond to elementary Petri nets is proven. It is shown how to generate all elementary Petri nets corresponding to a given transition system. If there is any such elementary Petri net, it is proven that there always exists a small one which has only polynomially many elements in the size of the transition system. Received October 5, 1992 / April 11, 1995     

Semantics for consistent activation in context-oriented systems

ContextContext-oriented programming languages provide dedicated programming abstractions to define behavioral adaptations and means to combine those adaptations dynamically according to sensed context changes. Some of these languages feature programming abstractions to explicitly define interaction dependencies among contexts. However, the semantics of context activation and the meaning of dependency relations have been described only informally, which in some cases has led to incorrect specifications, faulty implementations and inconsistent system behavior.ObjectiveWith the aim of avoiding faulty implementations and inconsistencies during system execution, this paper proposes both a formal and run-time model of contexts, context activation and context interaction.MethodAs a formal and computational basis, we introduce context Petri nets, a model based on Petri nets, which we found to match closely the structure of contexts in context-oriented systems. The operational semantics of Petri nets permits the modeling of run-time context activations. Existing Petri net analyses allow us to reason about system properties. As validation, we carried out small and medium-sized case studies.ResultsIn the cases explored, context Petri nets served effectively as underlying run-time model to ensure that declared context interaction constraints remain consistent during context manipulation. Moreover, context Petri nets enabled us to analyze certain properties regarding the activation state of particular contexts.ConclusionContext Petri nets thus proved to be appropriate to encode and manage the semantics of context activation, both formally and computationally, so as to preserve the consistency of context-oriented systems.