Petri Nets and Matrix Graph Grammars: Reachability

Net processes correspond to derivation processes in graph grammars

Graph Rewriting Systems are a powerful formalism for the specification of parallel and distributed systems, and the corresponding theory is rich of results concerning parallelism and concurrency. I will review the main results of the theory of concurrency for the algebraic approach to graph rewriting, emphasizing the relationship with the theory of Petri nets. In fact, graph rewriting systems can be regarded as a proper generalization of Petri nets, where the current state of a system is described by a graph instead of by a collection of tokens. Recently, this point of view allowed for the generalization to graph rewriting of some interesting results and constructions of the concurrent semantics of nets, including processes, unfoldings, and categorical semantics based on pair of adjoint functors.KeywordsDirect DerivationGraph TransformationGraph GrammarAdjoint FunctorHost GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the area of specification and modelling of concurrent systems, Petri nets have become a standard tool, and they still work behind the scenes in tools for graph transformation systems (cf. [1]). Moreover there is still potential for crossfertilization between the graph transformation and Petri net community. Even a better understanding of adhesive categories [2] and the related concepts of [3] seems possible in light of the proposed notion of weakly adhesive categories [4], which has emerged during work on the generalization of the co-reflective semantics of Petri nets to the realm of graph transformation and adhesive rewriting systems.KeywordsType ObjectGraph TransformationConcurrent SystemGraph GrammarGraph Transformation SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

Recently the general theory of graph grammars has become a growing area of research. Some properties which hold for all sequential, vertex-replacing graph grammars (without erasing) are presented, including a vertex pumping lemma. A construction is described which proves the undecidability of the question whether a graph grammar has the following property: changing the order of application of the productions in a derivation does not change the graph produced. Classes of graph grammars for which this property can be decided are presented. They include the NLC graph grammars of Janssens and Rozenberg [4,5].

Problems such as low transport efficiency, low safety factor, and difficulty in the dispatching of hybrid locomotives, which are not optimal in the automated transport of coal mines, are always difficult to study. These serious situations are studied and avoided through petri net (PN) analysis techniques in this paper. Firstly, by analyzing the relationship between locomotives, road sections and elements of petri net, petri net scheduling mode is built for locomotives with three different routes according to three different mine locomotive scheduling methods. Then, based on a certain set of resources in PN, the concepts of a deadlock state equation and a potential deadlock state equation are introduced. The deadlock state equation describes the intrinsic relationship between resources assignation and a deadlock state in PN. This equation is a necessary and sufficient condition for the occurrence of a deadlock situation. In the following part, the deadlock equations of several network models are solved, and the deadlock verification is carried out using the software PIPE, which proves the validity of the equations. The results show that the solution of the deadlock equation and the potential deadlock equation can effectively avoid the deadlock of petri net model, so as to prevent the blocking and collision of locomotives. At the same time, the whole modeling process also verifies the effectiveness of petri net for mining bottom locomotive modeling.

Firstly, net theory, which is a well established theory of distributed systems and processes [I] relies heavily on a graphical notation. One of the strengths of this theory is that many of the fundamental notions concerning concurrent systems and in particular, the dynamics of a concurrent systemcan be made transparent with the help of simple net diagrams and transformations of such diagrams. It is important to provide a solid formal foundation to this aspect of net theory and graph grammars appear to be the obvious tool for doing so. The second motivation is that in the study of concurrency problems that arise within the area of graph grammars [2] we think it is possible and fruitful to apply the concepts and techniques of net theory.

Petri nets have been proposed initially to model concurrency and distributed systems. The ordinary formalism is so simple with a smart formal background, but it fails in modeling mobility aspects. In spite of this, searchers in mobility modeling domain are attracted by Petri nets, and many extensions have been proposed to treat mobility aspects. Among these, we find Labeled Reconfigurable Nets (LRN), This new formalism allows explicit modeling of computational environments and processes mobility between them. It allows, in a simple and an intuitive approach, modeling mobile code paradigms (mobile agent, code on demand, remote evaluation). In this paper, we propose an approach based on the combined use of Meta-modeling and Graph Grammars to automatically generate a visual modeling tool for LRN, for analysis and simulation purposes. This tool produces graphical models, highly-structured to reach this goal. In our approach, the UML Class diagram formalism is used to define a meta-model of LRN. The meta-modeling tool ATOM3 is used to generate a visual modeling tool according to the proposed LRN meta-model. To generate R-Maude specification of the graphically specified LRN models, we have proposed a graph grammar. Then the reconfigurable rewriting logic language R-Maude can be used to perform the simulation of the resulted R-Maude specification. Our approach is illustrated through examples.

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.

Relating SPO and DPO graph rewriting with Petri nets having read, inhibitor and reset arcs

In the framework of graph transformation systems with Negative Application Conditions (NACs) the classical notion of switch equivalence of derivations is extended to permutation equivalence, because there are intuitively equivalent derivations which are not switch-equivalent if NACs are considered. By definition, two derivations are permutation-equivalent, if they respect the NACs and disregarding the NACs they are switch equivalent. A direct analysis of permutation equivalence is very complex in general, thus we propose a much more efficient analysis technique. For this purpose, we construct a Place/Transition Petri net, called dependency net, which encodes the dependencies among rule applications of the derivation, including the inhibiting effects of the NACs. The analysis of permutation equivalence is important for analysing simulation runs within development environments for systems modelled by graph transformation. The application of the technique is demonstrated by a graph transformation system within the context of workflow modelling. We show the effectiveness of the approach by comparing the minimal costs of a direct analysis with the costs of the efficient analysis applied to a derivation of our example system.

This article introduces a new way of generating a state equation which is useful for analyzing token flows in a Petri Net (PN). Transition values as used in the conventional state equation are replaced with transition variables, which are generated by multiplying a series of firing condition functions. The proposed state equation generalizes the conventional one in that it is represented with the transition variable form. A token flow with deadlock is analyzed as an example to establish the validity of the proposed state equation.

Hybrid continuous-time and discrete-event system models are developed for applications to intelligent robot and autonomous control systems. Certain general properties, including the interface structure between the discrete event part and the continuous part of hybrid systems, are studied. Petri nets are proposed as the model for the discrete event sub-system (DES), and state equations with a set of binary parameters as the model for the continuous time sub-system (CTS) in a hybrid system. The quantized CTS is formulated in prime event structures, which can also be modeled by a condition/event Petri net. Therefore, a complete hybrid system can be fit into an extended Petri net formalism, which provides a unified framework for hybrid system description. The validity of the proposed model is demonstrated by modeling a two-tank system.

Graph Grammars are a visual formal language for which there are several techniques of system specification and verification. This fact makes them an attractive option for describing complex systems. Fuzzy graph grammars generalize the concept of graph grammars, using fuzzy graphs. A fuzzy graph is composed of vertices and edges that have an associated membership value, which varies within a range of 0 to 1. Due to the lack of technique and analysis tools for fuzzy graph grammars, their use is still quite limited. One way to allow the analysis of fuzzy graphs grammars is using the extension of existing graph grammar approaches, including the concept of membership in the graph components. There is a relational approach of graph grammars that allows the use of theorem provers to perform property analysis of systems described in this language. This approach is the basis of a translation of graph grammars into the Event-B language, which enables the use of theorem provers of the Rodin tool. Therefore, the aim of this work is to extend this graph grammar approach, including fuzzy concepts, to allow the use of the theorem proving technique to analyze fuzzy graph grammars. In order to set this extension was first necessary to define the notion of typed fuzzy graph grammars.

Rewriting systems over adhesive categories have been recently introduced as a general framework which encompasses several rewriting-based computational formalisms, including various modelling frameworks for concurrent and distributed systems. Here we begin the development of a truly concurrent semantics for adhesive rewriting systems by defining the fundamental notion of process, well-known from Petri nets and graph grammars. The main result of the paper shows that processes capture the notion of true concurrency—there is a one-to-one correspondence between concurrent derivations, where the sequential order of independent steps is immaterial, and (isomorphism classes of) processes. We see this contribution as a step towards a general theory of true concurrency which specialises to the various concrete constructions found in the literature.KeywordsIsomorphism ClassDirect DerivationType ObjectGraph TransformationGraph GrammarThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.