Behavior Of Petri Nets Research Articles

Resource Bisimilarity in Petri Nets is Decidable

Petri nets are a popular formalism for modeling and analyzing distributed systems. Tokens in Petri net models can represent the control flow state or resources produced/consumed by transition firings. We define a resource as a part (a submultiset) of Petri net markings and call two resources equivalent when replacing one of them with another in any marking does not change the observable Petri net behavior. We consider resource similarity and resource bisimilarity, two congruent restrictions of bisimulation equivalence on Petri net markings. Previously it was proved that resource similarity (the largest congruence included in bisimulation equivalence) is undecidable. Here we present an algorithm for checking resource bisimilarity, thereby proving that this relation (the largest congruence included in bisimulation equivalence that is a bisimulation) is decidable. We also give an example of two resources in a Petri net that are similar but not bisimilar.

Testing Equivalences of Time Petri Nets

In the paper, we study a family of testing equivalences in interleaving, partial-order semantics, and combined semantics in the context of safe time Petri nets (elementary net systems whose transitions are labeled with time firing intervals, can fire only if their lower time bounds are attained, and are forced to fire when their upper time bounds are reached). For this purpose, the following three representations of behavior of safe time Petri nets are developed: sequences of firings of net transitions, which represent interleaving semantics; time causal processes, from which partial orders are derived; and time causal tree, whose nodes are sequences of transition firings and arcs are labeled by information about partial orders. We establish relationships between these equivalences and show that semantics of time causal processes and time causal trees coincide.

Full Axiomatisation of Timed Processes of Interval-Timed Petri Nets

In this paper we use partial order semantics to express the truly concurrent behaviour of Interval-Timed Petri nets (ITPNs) in their most general setting, i.e. with autoconcurrency and zero duration, as studied with its standard maximal step semantics in [1]. First we introduce the notion of timed processes for ITPNs inductively. Then we investigate if the equivalence of inductive and axiomatic process semantics - true for classical Petri nets - could hold for ITPNs too. We will see that the notions of independence and immediate firing obligation seem to be antagonistic ones, and that local axioms, adequate to define processes of classical Petri nets, are not sufficient to caracterize timed processes of ITPNs. We propose several original “global” axioms which reveal to be an effective solution. Thus we yield finally a full axiomatic definition of timed processes for ITPNs.

Controlling Petri Net Behavior using Priorities for Transitions

In this paper we examine how it is possible to control Petri net behavior with the help of transition priorities. Controlling here means forcing a process to behave in a stable way by ascribing priorities to transitions and hence transforming a classic Petri net into a Priority Petri net. For Petri net models stability is often ensured by liveness and boundedness. These properties are crucial in many application areas, e.g. workflow modeling, embedded systems design, and bioin- formatics. In this paper we study the problem of transforming a given live, but unbounded Petri net into a live and bounded one by adding priority constraints. We specify necessary conditions for the solvability of this problem and present an algorithm for ascribing priorities to net transitions in such a way that the resulting net becomes bounded while staying live.

Implementing a Matrix Vector Transition Net

Aims: Classic Petri nets also known as place transition nets provide many interesting and useful features for system modeling. However they are limited by the place types that are used. An approach is presented where a matrix vector transition net is used for modeling complex system behavior. This is more expressive than a Petri net. Proposed Solution: A modified form of a traditional place transition net is used. The idea is to combine the ideas from Petri net behavior with the matrix vector approach. Implementing the Matrix Vector Transition Net: Ordinary Petri net places are replaced with matrices or vectors. The input and output arcs have a specific function matrix that determines firing. Firing and behavior remains conceptually similar to that of a petri net. Row and column vectors can also be used interchangeably. The behavior of matrix transition nets is similar to that of a place transition net. Instead of tokens, matrix elements are used. This type of structure increases the modeling power, abstraction capacity and the complexity of the net. Case Study: A toy case of an abstract network structure containing processing elements is used to illustrate the use of the matrix vector transition net structure. Results and Findings: The behavior of matrix transition nets is shown to be similar to that of a place transition net. Instead of tokens, matrix elements are used. It is possible to construct a symbolic marking graph or reachability graph for the system This type of structure definitely increases the modeling power, abstraction capacity and the complexity of the net. The matrix transition net could be useful for certain types of communication system problems and complex system interfacing.

Necessary and sufficient liveness condition of GS3PR Petri nets

Structural analysis is one of the most important and efficient methods to investigate the behaviour of Petri nets. Liveness is a significant behavioural property of Petri nets. Siphons, as structural objects of a Petri net, are closely related to its liveness. Many deadlock control policies for flexible manufacturing systems (FMS) modelled by Petri nets are implemented via siphon control. Most of the existing methods design liveness-enforcing supervisors by adding control places for siphons based on their controllability conditions. To compute a liveness-enforcing supervisor with as much as permissive behaviour, it is both theoretically and practically significant to find an exact controllability condition for siphons. However, the existing conditions, max, max′, and max″-controllability of siphons are all overly restrictive and generally sufficient only. This paper develops a new condition called max*-controllability of the siphons in generalised systems of simple sequential processes with resources (GS3PR), which are a net subclass that can model many real-world automated manufacturing systems. We show that a GS3PR is live if all its strict minimal siphons (SMS) are max*-controlled. Compared with the existing conditions, i.e., max-, max′-, and max″-controllability of siphons, max*-controllability of the SMS is not only sufficient but also necessary. An example is used to illustrate the proposed method.

Building Occurrence Nets from Reveals Relations

Occurrence nets are a well known partial order model for the concurrent behavior of Petri nets. The causality and conflict relations between events, which are explicitly represented in occurrence nets, induce logical dependencies between event occurrences: the occurrence of an event e in a run implies that all its causal predecessors also occur, and that no event in conflict with e occurs. But these structural relations do not express all the logical dependencies between event occurrences in maximal runs: in particular, the occurrence of e in any maximal run may imply the occurrence of another event that is not a causal predecessor of e, in that run. The reveals relation has been introduced to express this dependency between two events. Here we generalize the reveals relation to express more general dependencies, involving more than two events, and we introduce ERL logic to express them as boolean formulas. Finally we answer the synthesis problem that arises: given an ERL formula Φ, is there an occurrence net N such that Φ describes exactly the dependencies between the events of N?

Branching Processes of General Petri Nets

We propose a new model of branching processes, suitable for describing the behavior of general Petri nets, without any finiteness or safeness assumption. In this framework, we define a new class of branching processes and unfoldings of a net N, which we call faithful. These coincide with the safe branching processes and unfoldings if N is safe, or weakly safe as in [Engelfriet 1991], but not in general. However, faithful branching processes and processes satisfy the good order-theoretic properties which make the safe processes of safe nets so useful in practice, and which are known not to hold for the safe processes of a general net. Faithful processes represent therefore good candidates to generalize the theory of safe nets to the general case.

Synthesis of Nets with Step Firing Policies

The unconstrained step semantics of Petri nets is impractical for simulating and modelling applications. In the past, this inadequacy has been alleviated by introducing various flavours of maximally concurrent semantics, as well as priority orders. In this paper, we introduce a general way of controlling step semantics of Petri nets through step firing policies that restrict the concurrent behaviour of Petri nets and so improve their execution and modelling features. In a nutshell, a step firing policy disables at each marking a subset of enabled steps which could otherwise be executed. We discuss various examples of step firing policies and then investigate the synthesis problem for Petri nets controlled by such policies. Using generalised regions of step transition systems, we provide an axiomatic characterisation of those transition systems which can be realised as reachability graphs of Petri nets controlled by a given step firing policy. We also provide two different decision and synthesis algorithms for PT-nets and step firing policies based on linear rewards of steps, where the reward for firing a single transition is either fixed or it depends on the current net marking. The simplicity of the algorithms supports our claim that the proposed approach is practical.

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].

Dealing with Mutual Exclusion Sections in Production Systems: from Shared Resources to Parallel TEG's

This paper deals with the disambiguation of the behaviour of Petri nets including shared resources. In the production management context, they are often used for the modelling of manufacturing cells. But this representation has a poor transposition into dioid algebra. In this article, we design a method to describe such a phenomenon in a dioid of interval. The latter expresses this class of Petri nets models in a formal way. Their input/output behaviours are guaranteed to be greater than the lower bound of the reference model set and lower than the upper bound of this set. In fact, the resource sharing problem is turned into a time uncertainty problem, concerning the access to the shared resource. In this new problem, time uncertainties are bounded and can be described by intervals.Both bounds “confining” the behaviours of the studied production systems in intervals can be manipulated in the scope of the &ℤbar;max algebra, even though the original systems are not &ℤbar;max linear by essence.

The tool TINA – Construction of abstract state spaces for petri nets and time petri nets

In addition to the graphic-editing facilities, the software tool Tina proposes the construction of a number of representations for the behaviour of Petri nets or Time Petri nets. Various techniques are used to extract views of the behaviour of nets, preserving certain classes of properties of their state spaces. For Petri nets, these abstractions help prevent combinatorial explosion, relying on so-called partial order techniques such as covering steps and/or persistent sets. For Time Petri nets, which have, in general, infinite state spaces, they provide a finite symbolic representation of their behaviour in terms of state classes.

Clustering for Petri nets

This work builds a bridge from • clustering techniques—merging neighbouring nodes which is a key feature for software engineering and the practical applications of Petri nets—to • folding techniques—merging only transitions with transitions and places with places, preserving behaviour and allowing theoretical connections to many models of concurrency. A new category of Petri nets is introduced. Morphisms support clustering, offering attractive properties to software engineering and integrating smoothly with invariants. A computationally reasonable adjunction connects it to folding-based Petri nets, namely, to two new cocomplete and complete categories. The dichotomy of structure and behaviour of Petri nets is expressed as compatible adjunctions to behavioural categories. Finally reachability and process semantics are attached categorically and a new variant of occurrence nets is proposed as a purer image of causality and branching. This framework offers categorical support for practical applications of Petri nets.

A HYBRID APPROACH TO KNOWLEDGE ACQUISITION USING NEURAL PETRI NETS AND DS THEORY

Knowledge acquisition from multiple experts and its refinement are important issues in knowledge management of an expert system. The paper presents a novel approach to handling the above problems by combining the synergistic behavior of neural Petri nets and the Dempster–Shafer theory. The Dempster–Shafer theory has been employed here to reduce the scope of uncertainty in the supplied noisy input instances and the inferences generated therefrom by the multiple experts. The noise-free training instances thus obtained are subsequently used to train the neural Petri net model for refining the parameters of its knowledge base. A comparison of the performance of the proposed training algorithm with the classical back-propagation algorithm has also been presented in the paper.

Modeling and analysis of timed Petri nets using heaps of pieces

The authors show that safe timed Petri nets can be represented by special automata over the (max, +) semiring, which compute the height of heaps of pieces. This extends to the timed case the classical representation a la Mazurkiewicz of the behavior of safe Petri nets by trace monoids and trace languages. For a subclass including all safe free-choice Petri nets, we obtain reduced heap realizations using structural properties of the net (covering by safe state machine components). The authors illustrate the heap-based modeling by the typical case of safe jobshops. For a periodic schedule, the authors obtain a heap-based throughput formula, which is simpler to compute than its traditional timed event graph version, particularly if one is interested in the successive evaluation of a large number of possible schedules.

Symbolic computation tree logic model checking of time Petri nets

Model checking is a technique to decide whether a system modeled by a state graph satisfies a temporal logic formula. A serious problem in model checking of practical systems is that the size of the state graphs becomes huge. One solution of this problem is symbolic model checking, in which sets of states and transition relations over states are represented and handled symbolically. This research studies symbolic model checking in which realtime systems are modeled by time Petri nets and specifications are expressed by computation tree logic formulas. As realtime systems potentially have infinite state spaces, techniques to reduce infinite state spaces to finite ones are necessary. This paper discusses a technique based on equivalence classes, called regions, of states as well as a technique to restrict the behavior of time Petri nets so that transitions fire only at integer times. The algorithm in the former approach is presented in detail and then some experimental results are shown. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(4): 11–20, 1997

An axiomatization of the algebra of Petri net concatenable processes

The concatenable processes of a Petri net N can be characterized abstractly as the arrows of a symmetric monoidal category P ( N ). However, this is only a partial axiomatization, since it is based on a concrete, ad hoc chosen, category of symmetries Sym N . In this paper we give a completely abstract characterization of the category of concatenable processes of N , thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets.

An axiomatization of the algebra of Petri net concatenable processes

The concatenable processes of a Petri net N can be characterized abstractly as the arrows of a symmetric monoidal category P ( N). However, this is only a partial axiomatization, since it is based on a concrete, ad hoc chosen, category of symmetries Sym N . In this paper we give a completely abstract characterization of the category of concatenable processes of N, thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets.

An event structure semantics for general Petri nets

In this paper we address the following question: What type of event structures are suitable for representing the behaviour of general Petri nets? As a partial answer to this question we define a new class of event structures called local event structures and identify a subclass called UL-event structures. We propose that UL-event structures are appropriate for capturing the behaviour of general Petri nets. Our answer is a partial one in that in the proposed event structure semantics, auto-concurrency is filtered out from the behaviour of Petri nets. It turns out that this limited event structure semantics for Petri nets is nevertheless a non-trivial and conservative extension of the (prime) event structure semantics of 1-safe Petri nets provided in Nielsen et al. (1981). We also show that the strong relationship between prime event structures and 1-safe Petri nets established in a categorical framework in Winskel (1987) can be extended to the present setting, provided we restrict our attention to the subclass of Petri nets whose behaviours do not exhibit any auto-concurrency. Finally, we show that Winskel's general and stable event structures can be smoothly related to local event structures and that similarly prime event structures can be related to UL-event structures.

Process versus unfolding semantics for Place/Transition Petri nets

In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical “token game”, one can model the behaviour of Petri nets via nonsequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. In our formal development a relevant role is played by DecOcc , a category of occurrence nets appropriately decorated to take into account the history of tokens. The structure of decorated occurrence nets at the same time provides natural unfoldings for Place/Transition (PT) nets and suggests a new notion of processes, the decorated processes, which induce on Petri nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification.