Concurrent graph and term graph rewriting

Net processes correspond to derivation processes in graph grammars

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.

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.

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.

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.

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.

The PETRI NETS 2023 proceedings deal with the application and theory of petri nets, distributed and concurrency systems, etc.

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.

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.

In the construction of a software system, software documents (requirements definitions, software architectures, module implementations, etc.) are produced which are written in different languages and describe the system from different points of view. The construction of tools for the integration of such documents is addressed. A formal approach to the semi-automatic construction of incremental integration tools is presented. Passive and active support for interdocument dependencies (consistency analyses and generation/updating of templates for dependent documents, respectively) are provided. For formally specifying incremental integration tools, graph rewriting systems are used. It is shown that certain parts of such a graph rewriting system may be generated from a simple description of language correspondences. >

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.

Attendees of the workshop, especially those coming from the very applied side, might ask themselves why it is interesting to consider the workshop from the theoretical perspective. A workshop which is clearly dedicated to applications of graph transformation concepts! Let us have a short look back to research on graph transformation about ten years ago. In the beginning of the nineties, the research community on graph transformation worked mainly on theoretical problems. Nowadays, graph transformation is applied in very different fields and the amount of practical work increased considerably. Compare e.g. the contributions to the 1. International Conference on Graph Transformation last year in Barcelona [1]. So the question arises: What is the significance of theoretical work on graph transformation nowadays?

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.

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.