Petri nets and their relation to graph grammars

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.

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

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.

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.

The new approach of distributed graphs and graph transformation developed in this article allows us to use structured graph transformation on two abstraction levels: the network and the local level. The network level contains the description of topological structures of a system. The local level covers the description of states and their transitions in local systems. Local state transitions may depend on others using suitable synchronization mechanisms. The main distribution concepts of categorical graph grammars presented by Schneider are combined with the algebraic approach to distributed graph transformation introduced by Ehrig et al. Modeling of distributed systems by this new kind of distributed graph transformation offers a clear and elegant description of dynamic networks, distributed actions as well as communication and synchronization using a graphical notation. Moreover, distributed graph transformation offers the possibility to describe splitting and joining of local graphs as well as parallel transformations in local systems. The formalization of distributed graph transformation is performed by means of category theory. A distributed graph is formalized by a diagram in the category GRAPH of graphs and total graph morphisms. A distributed transformation step is characterized by a double-pushout in the category DISTR(GRAPH) of distributed graphs and distributed graph morphisms. A pushout over distributed graph morphisms cannot always be constructed componentwise in each local part. But especially the componentwise construction of a distributed transformation reflects distributed computations best. Thus, we present the necessary conditions for componentwise construction of pushouts in DISTR(GRAPH) and show that they are also sufficient. These conditions are summarized in the distributed gluing condition. Moreover, these conditions are needed for componentwise construction of pushout complements which are used to characterize the first step of a distributed graph transformation.

Petri net paradigm of computing is concerned discussing basic types of Petri net computers and their relations. We developed a technique of parametric specification of Petri nets, built of a few connected components which are repeated depending on a parameter or a set of parameters, and composition of software to generate automatically the corresponding models in a vivid graphical form. We construct explicitly a Petri net that is a strong computer of double exponent after R.J. Lipton. Its nondeterministic computation is represented by a Petri net reachability graph to which a successful branch belongs. Then we minimise and transform the net to obtain twice smaller construct than the original one. Moreover, the obtained net possesses such a marvellous property that addingto it four priority arcs makes it an exact (deterministic) computer having a single firable sequence only. Same as the Lipton’s net, the minimized net allows the proof that the Petri net reachability problem is exponential in space. Besides, it represents an example when simple transformations allow guessing the successful branch of a nondeterministic computation. The practical significance of the approach in applications consists in a state-of-art heuristic technique of solving some NP problems in polynomial time.

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.

One of the objections to the use of Petri nets or, equivalently, specifications written in the COSY (Concurrent Systems) notation, for modelling realistic systems is that they would grow too large to be of any practical use. Generators for the concise representation of large (possibly infinite) structures in net theory or specifications in COSY are traced to their origins in Carl Adam Petri’s thesis [P62]. The generators implemented in the current version of COSY and its accompanying simulation and analysis tools are presented in detail together with design decisions which led to their present form.

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.

A Petri net can provide a graphical model of a concurrent system. They have been used to model such varied systems as resource control, communication networks and manufacturing processes. Computer Science students will, hopefully, learn the theory of Petri nets in a discrete mathematics course or a course on theoretical foundations of Computer Science. They may even see a graph of a sample net in a textbook, but it is unlikely they will have a chance to experiment with a live net. The Interactive Petri Net Simulator (IPNetS) described in this paper provides an easy point-and-click environment for students to design, modify, examine, and execute a Petri net. In seconds any simple Petri net can be created and animated. IPNetS provides display, file handling, and user control functions while a separate execution unit does the actual execution of the net. This flexible design allows for easy experimentation with different execution units and different types of nets. An Ada and a C++ execution unit are contrasted.

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 aim of this work is the modeling and verification of concurrent systems subject to dynamic changes using extensions of Petri nets. We begin by introducing the notion of net rewriting system. In a net rewriting system, a system configuration is described as a Petri net and a change in configuration is described as a graph rewriting rule. We show that net rewriting systems are Turing powerful, that is, the basic decidable properties of Petri nets are lost and, thus, automatic verification in not possible for this class. A subclass of net rewriting systems are reconfigurable Petri nets. In a reconfigurable Petri net, a change in configuration amounts to the modification of the flow relations of the places in the domain of the involved rule according to this rule, independently of the context in which this rewriting applies. We show that reconfigurable Petri nets are formally equivalent to Petri nets. This equivalence ensures that all the fundamental properties of Petri nets are still decidable for reconfigurable Petri nets and this model is thus amenable to automatic verification tools. Therefore, the expressiveness of both models is the same, but, with reconfigurable Petri nets, we can easily and directly model systems that change their structure dynamically.