Net processes correspond to derivation processes in graph grammars

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].

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.

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.

The aim of this paper is to give an introduction how to use categorical methods in a specific field of computer science: The field of high-level-replacement systems has its roots in the well-established theories of formal languages, term rewriting, Petri nets, and graph grammars playing a fundamental role in computer science. More precisely, it is a generalization of the algebraic approach to graph grammars which is based on gluing constructions for graphs defined as pushouts in the category of graphs. The categorical theory of high-level-replacement systems is suitable for the dynamic handling of a large variety of high-level structures in computer science including different kinds of graphs and algebraic specifications. In this paper we discuss the basic principles and techniques from category theory applied in the field of high-level-replacement systems and present some basic results together with the corresponding categorical proof techniques.

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.

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.

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.

Motivated by the work which has been done for Petri-nets, the paper presents a categorical approach to graph grammars “in the large”. In the large means, that we define categories of graph grammars, graph transition systems, and graph derivation systems which embody the notion “grammar”, “direct derivation”, and “derivation”, respectively, as they are defined in the classical algebraic theory. For this purpose we introduce a suitable notion of graph grammar morphism on “typed graph grammars” in analogy to Petri-nets. A typed graph grammar is a grammar for typed graphs which is a slight generalization of the standard case. The main result shows that the three categories are related by left-adjoint functors. We discuss the relationship of our results to similar results obtained in the Petri-net field, and applications to entity/relationship models.

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.

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.

In this paper we present algebraic high-level nets: a combination of algebraic specifications and Petri nets. Algebraic specifications are used to specify the data-structure and Petri nets to specify the data-flow. This combination is a very powerful description technique. Moreover we introduce net transformations, net fusions and net unions as structuring techniques. Fusions and unions can be considered as horizontal structuring mechanisms in the sense that we combine nets to obtain a new net consisting of the given components. Fusions capture the notion of sharing of subnets, while unions are kinds of net composition. Net transformations are based on concepts from graph grammars because net refinements are defined via productions. Transformations can be seen as high-level vertical refinements. Moreover we show compatibility between these techniques, i.e. in our framework vertical refinement and horizontal structuring of nets are compatible.

The problem of extending to graph grammars the unfolding semantics originally developed by Winskel for (safe) Petri nets has been faced several times along the years, both for the single-pushout and double-pushout approaches, but only partial results were obtained. In this paper we fully extend Winskel’s approach to single-pushout grammars providing them with a categorical concurrent semantics expressed as a coreflection between the category of graph grammars and the category of prime algebraic domains.

This paper attempts to contribute in two directions. First, concepts and results of our Matrix Graph Grammars approach [VL06b, VL06a] such as coherence and minimal initial digraph are applied to Petri nets, especially to reachability criteria and to the state equation. Second, the state equation and related Petri nets techniques for reachability are generalized to cover a wider class of graph grammars.

Several formal concurrent semantics have been proposed for graph rewriting, a powerful formalism for the specification of concurrent and distributed systems which generalizes P/T Petri nets. In this paper we relate two such semantics recently proposed for the algebraic double-pushout approach to graph rewriting, namely the derivation trace and the graph process semantics. The notion of concatenable graph process is introduced and then the category of concatenable derivation traces is shown to be isomorphic to the category of concatenable graph processes. As an outcome we obtain a quite intuitive characterization of events and configurations of the event structure associated to a graph grammar.