Some decision problems related to the reachability problem for Petri nets

In this paper, we show that (1) the question to decide whether a given Petri net is consistent, M_o-reversible or live is reduced to the reachability problem in a unified manner, (2) the reachability problem for Petri nets is equivalent to the equality problem and the inclusion problem for the sets of all firing sequences of two Petri nets, (3) the equality problem for the sets of firing sequences of two Petri nets with only two unbounded places under homomorphism is undecidable, (4) the coverability and reachability problems are undecidable for generalized Petri nets in which a distinguished transition has priority over the other transitions, and (5) the reachability problem is undecidable for generalized Petri nets in which some transitions can reset a certain place to zero marking.     

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.     

Deciding a class of path formulas for conflict-free petri nets

The aim of this paper is to develop a unified approach for deriving complexity results for problems concerning conflict-free Petri nets. To do so, we first define a class of formulas for paths in Petri nets. We then show that answering the satisfiability problem for conflict-free Petri nets is tantamount to solving a system of linear inequalities (which is known to be in P). Since a wide spectrum of Petri net problems (including various fairness-related problems) can be reduced to the satisfiability problem in a straightforward manner, our approach offers an umbrella under which many Petri net problems for conflict-free Petri nets can be shown to be solvable in polynomial time. As a side-product, our analysis provides evidence as to why detecting unboundedness for conflict-free Petri nets is easier (provided P ≠ NP) than for normal and sinkless Petri nets (which are two classes that properly contain conflict-free Petri nets). A preliminary version was presented at the 14th International Conference on Application and Theory of Petri Nets, Chicago, IL, USA, June 1993.     

State equation, controllability, and maximal matchings of petri nets

Petri nets are a versatile modeling device for studying the structure and control of concurrent systems. Petri nets and related graph models have been used for modeling a wide variety of systems from computers to social systems. In order to introduce this interesting modeling device to the researcher in control theory, this paper discusses Petri nets in the context of the state equation for a linear discrete-time system. The controllability concept of dynamic systems is applied to Petri nets for the first time. It is also shown that the controllability and reachability of a Petri net are related to maximal matchings of its bipartite graph.     

Detection of Ada static deadlocks using Petri net invariants

A method is presented for detecting deadlocks in Ada tasking programs using structural; and dynamic analysis of Petri nets. Algorithmic translation of the Ada programs into Petri nets which preserve control-flow and message-flow properties is described. Properties of these Petri nets are discussed, and algorithms are given to analyze the nets to obtain information about static deadlocks that can occur in the original programs. Petri net invariants are used by the algorithms to reduce the time and space complexities associated with dynamic Petri net analysis (i.e. reachability graph generation)     

无界Petri网的可达树的综述

Petri网自提出以来得到了学术界和工业界的广泛关注. Petri网系统的可达性是最基本性质之一.系统的其他相关性质都可以通过可达性进行分析.利用等价的有限可达树来研究无界Petri网可达性,依然是一个开放性问题.该研究可以追溯到40年前,但由于问题本身的复杂性和难度太大,直到最近20年,经过国内外诸多学者的不懈努力,才逐渐取得了一些阶段性的成果和部分突破.本文回顾了近40年来国内外学者为彻底解决该问题作出的贡献.重点对4种开创性的研究成果展开讨论,分别为有限可达树、扩展可达树、改进可达树及新型改进可达树.探讨了今后无界Petri网可达性问题的研究方向.     

Action Planning for Directed Model Checking of Petri Nets

Petri nets are fundamental to the analysis of distributed systems especially infinite-state systems. Finding a particular marking corresponding to a property violation in Petri nets can be reduced to exploring a state space induced by the set of reachable markings. Typical exploration(reachability analysis) approaches are undirected and do not take into account any knowledge about the structure of the Petri net. This paper proposes heuristic search for enhanced exploration to accelerate the search. For different needs in the system development process, we distinguish between different sorts of estimates.Treating the firing of a transition as an action applied to a set of predicates induced by the Petri net structure and markings, the reachability analysis can be reduced to finding a plan to an AI planning problem. Having such a reduction broadens the horizons for the application of AI heuristic search planning technology. In this paper we discuss the transformations schemes to encode Petri nets into PDDL. We show a concise encoding of general place-transition nets in Level 2 PDDL2.2, and a specification for bounded place-transition nets in ADL/STRIPS. Initial experiments with an existing planner are presented.     

Supervision Based on Place Invariants: A Survey

The supervision based on place invariants (SBPI) is an efficient technique for the supervisory control of Petri nets. This paper reveals the significance of the SBPI based on a literature survey, applications, and an analysis of problems and supervisory settings that can be addressed using SBPI. Special attention is given to the various settings within which the problem can be formulated. Such settings can be distinguished based on the concurrency type, the type of controllability and observability, and the centralized or decentralized type of supervision. As we show, it is possible to approach the most general settings in a purely structural way, without resorting to reachability analysis. We begin by describing the SBPI problem and the literature methods that address this problem or are related to it. Then, we proceed to show classes of problems that can be reduced to the SBPI problem. In the SBPI, the specification is described as a system of inequalities that the Petri net marking must satisfy at any time. However, as we show, problems involving more general specifications can be approached in the SBPI setting, sometimes under additional assumptions, by performing net and/or specification transformations. Four of the specifications we will consider are logic constraints, language specifications, disjunctions of linear constraints, and liveness. We conclude with a presentation of possible applications of the SBPI approach to programming with semaphores, fault tolerance, and synchronic-distance based designs.     

Verifying time Petri nets by linear programming

The paper proposes an approach to solving some verification problems of time petri nets using linear programming.The approach is based on the observation that for loop-closed time Ptri nets,it is only necessary to investigate a finite prefix of an untimed run of the underlying Petri net.Using the technique the paper gives solutions to reachabiltiy and bounded delay timing analysis problems.For both problems algorithms are given,that are decision procedures for loop-closed time Petri nets.and semi-decision procedures for general time Petri nets.     

An algebraic approach to revising propositional rule-based knowledge bases

One of the important topics in knowledge base revision is to introduce an efficient implementation algorithm. Algebraic approaches have good characteristics and implementation method; they may be a choice to solve the problem. An algebraic approach is presented to revise propositional rule-based knowledge bases in this paper. A way is firstly introduced to transform a propositional rule-based knowledge base into a Petri net. A knowledge base is represented by a Petri net, and facts are represented by the initial marking. Thus, the consistency check of a knowledge base is equivalent to the reachability problem of Petri nets. The reachability of Petri nets can be decided by whether the state equation has a solution; hence the consistency check can also be implemented by algebraic approach. Furthermore, algorithms are introduced to revise a propositional rule-based knowledge base, as well as extended logic programming. Compared with related works, the algorithms presented in the paper are efficient, and the time complexities of these algorithms are polynomial.     

Normal Petri nets

A class of Petri nets, called normal Petri nets, is introduced, and it is shown that, for each initial marking, the reachability set of a normal marked Petri net is an effectively computable semilinear set. More generally, we show that the reachability set of a marked Petri net is an effectively computable semilinear set unless the total number of tokens in a minimal circuit is decreased to 0. We also show that a Petri net is normal if and only if it is weakly persistent for each initial marking without token-free circuits.     

Hierarchical reachability graph of bounded Petri nets forconcurrent-software analysis

Petri nets have been proposed as a promising tool for modeling and analyzing concurrent-software systems such as Ada programs and communication protocol software. Among analysis techniques available for Petri nets, the most general approach is to generate all possible states (markings) of the system in a form of a so-called reachability graph. However, this conventional reachability graph approach is inefficient or intractable, even for a bounded Petri net, due to state explosion in many practical applications. To cope with this problem, this paper proposes a method for constructing a hierarchically organized state space called the hierarchical reachability graph (HRG). Using the HRG, we obtain necessary and sufficient conditions for reachability and deadlock, as well as algorithms to test whether a given state or marking is reachable from the initial state and whether there is a deadlock state (a state with no successor states)     

A Sufficient Condition for Reachability in a General Petri Net

Necessary and sufficient conditions for reachability exist only for special classes of Petri nets, acyclic Petri nets being one among them. We present a net transformation procedure that converts a general Petri net into an acyclic Petri net to utilize the available sufficient condition. We show the relationship between the reachable markings of the original Petri net and the associated acyclic Petri net. Given a firing count vector, we discuss how the sufficient condition for reachability in an acyclic Petri net could be utilized for a general Petri net. We also discuss the utility of the acyclic transformed net in determining the transition firing sequences of the reachable markings with known firing count vectors.     

Distributing Finite Automata Through Petri Net Synthesis

The synthesis problem for Petri nets consists in deciding constructively the existence of a Petri net with sequential state graph isomorphic to a given graph. If events are attached to locations, one may set as an additional requirement that the synthesised net should be distributable; i.e. such that events at different locations have no common input place, whence distributed conflicts are avoided. Distributable nets are easily implemented by finite families of automata (one per location) communicating with each other by asynchronous message passing. We show that the general Petri net synthesis problem and its distributed version may both be solved in time polynomial in the size of the given graph. We report on some preliminary experiments of Petri net synthesis applied to the distribution of reactive automata using the tool SYNET. Received November 2000 / Accepted in revised form August 2001     

Executability of scenarios in Petri nets

In this paper, we show that it can be tested in polynomial time as to whether a scenario is an execution of a Petri net. This holds for a wide variety of Petri net classes, ranging from elementary nets to general inhibitor nets. Scenarios are given by causal structures expressing causal dependencies and concurrency among events. In the case of elementary nets and of place/transition nets, such causal structures are partial orders among transition occurrences. For several extended Petri net classes, the extension of partial orders to stratified order structures is considered.The algorithms are based on the representation of the non-sequential behavior of Petri nets by so-called token flow functions and a characterization of Petri net executions called token flow property. This property allows nontrivial transformations into flow optimization problems, which can be solved in polynomial time. The paper is a revised, consolidated and extended version of the conference papers [G. Juhás, R. Lorenz, J. Desel, Can I execute my scenario in your net?, in: G. Ciardo, P. Darondeau (Eds.), ICATPN, in: Lecture Notes in Computer Science, Springer, 2005, pp. 289–308; R. Lorenz, S. Mauser, R. Bergenthum, Testing the Executability of Scenarios in General Inhibitor Nets, in: ACSD, IEEE Computer Society, 2007, pp. 167–176] and includes parts of the habilitation thesis [R. Lorenz, Szenario-basierte Verifikation und Synthese von Perinetzen: Theorie und Anwendungen, Habilitation, 2006].     

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.     

Structuring Acyclic Petri Nets for Reachability Analysis and Control

The incidence matrices—from places to transitions and vice versa—of an acyclic Petri net can obtain a block-triangular structure by reordering their rows and columns. This allows the efficient solution of some reachability problems for acyclic Petri nets. This result is further used in supervisory control of Petri nets; supervisors for Petri nets with uncontrollable transitions are constructed by extending the method of Yamalidou et al. (1996) to Petri nets where transitions can be executed simultaneously. A large class of Petri nets with uncontrollable transitions is given for which the maximally permissive supervisor can be realized by a Petri net. The original specification is algorithmically transformed—by using the results for acyclic Petri nets—into a new specification to take the presence of uncontrollable transitions into account. The supervisor is obtained by simple matrix multiplications and no linear integer programs need to be solved. Furthermore, a class of Petri nets is given for which the supervisor can be realized by extending the enabling rule with OR-logic.     

State space analysis of Petri nets with relation-algebraic methods

A large variety of systems can be modelled by Petri nets. Their formal semantics are based on linear algebra which in particular allows the calculation of a Petri net’s state space. Since state space explosion is still a serious problem, efficiently calculating, representing, and analysing the state space is mandatory. We propose a formal semantics of Petri nets based on executable relation-algebraic specifications. Thereupon, we suggest how to calculate the markings reachable from a given one simultaneously. We provide an efficient representation of reachability graphs and show in a correct-by-construction approach how to efficiently analyse their properties. Therewith we cover two aspects: modelling and model checking systems by means of one and the same logic-based approach. On a practical side, we explore the power and limits of relation-algebraic concepts for concurrent system analysis.     

Supervisory control of deterministic Petri nets with regularspecification languages

Algorithms for computing a minimally restrictive control in the context of supervisory control of discrete-event systems have been well developed when both the plant and the desired behaviour are given as regular languages. In this paper the authors extend such prior results by presenting an algorithm for computing a minimally restrictive control when the plant behaviour is a deterministic Petri net language and the desired behaviour is a regular language. As part of the development of the algorithm, the authors establish the following results that are of independent interest: i) the problem of determining whether a given deterministic Petri net language is controllable with respect to another deterministic Petri net language is reducible to a reachability problem of Petri nets and ii) the problem of synthesizing the minimally restrictive supervisor so that the controlled system generates the supremal controllable sublanguage is reducible to a forbidden marking problem. In particular, the authors can directly identify the set of forbidden markings without having to construct any reachability tree     

Petri网的符号ZBDD可达树分析技术

Petri网是一种适合于并发系统建模、分析和控制的图形工具.可达树是Petri网分析的典型技术之一,它通过标识向量集合表征系统的状态空间,组合复杂性严重制约了该分析技术可处理系统问题的规模.零压缩决策图(Zero-Suppressed Binary Decision Diagrams,ZBDD)是一种新型的数据结构,是表示和处理稀疏向量集合的一种有效技术.文章基于Petri网町达标识向量的稀疏特征,给出了Petri网分析的符号ZBDD技术,该技术通过对标识向量(状态)的布尔向量表示、可达标识向最(状态)的符号ZBDD生成,实现Petri网可达状态空间的高效符号操作和紧凑符号表示.实验表明,基于ZBDD的符号可达性分析算法能够有效处理较大规模Petri网问题.