In this paper we introduce a static analysis technique for graph transformation systems. We present an algorithm which, given a graph transformation system and a start graph, produces a nite structure consisting of a hypergraph decorated with transitions (Petri graph) which can be seen as an approximation of the Winskel style unfolding of the graph transformation system. The fact that any reachable graph has an homomorphic image in the Petri graph and the additional causal information provided by transitions allow us to prove several interesting properties of the original system. As an application of the proposed technique we show how it can be used to verify the absence of deadlocks in an innite-state Dining Philosophers system.

It is well-known that a term rewriting system can be faithfully described by a cartesian 2-category, where horizontal arrows represent terms, and cells represent rewriting sequences. In this paper we propose a similar, original 2-categorical presentation for term graph rewriting. Building on a result presented in [8], which shows that term graphs over a given signature are in one-to-one correspondence with arrows of a gs-monoidal category freely generated from the signature, we associate with a term graph rewriting system a gs-monoidal 2-category, and show that cells faithfully represent its rewriting sequences. We exploit the categorical framework to relate term graph rewriting and term rewriting, since gs-monoidal (2-)categories can be regarded as "weak" cartesian (2-)categories, where certain (2-)naturality axioms have been dropped.

We propose an unfolding semantics for graph transformation systems in the double-pushout (DPO) approach. Mimicking Winskel’s construction for Petri nets, a graph grammar is unfolded into an acyclic branching structure, that is itself a (nondeterministic occurrence) graph grammar describing all the possible computations of the original grammar. The unfolding can be abstracted naturally to a prime algebraic domain and then to an event structure semantics. We show that such event structure coincides both with the one defined by Corradini et al. [3] via a

We propose an event based semantics for contextual nets, i.e. an extension of Place/Transition Petri nets where transitions can also have context conditions, modelling resources that can be read without being consumed. The result is a generalization of Winskel’s work on safe nets: the event based semantics is given at categorical level via a chain of coreflections leading from the category WS-CN of weakly safe contextual nets to the category Dom of finitary prime algebraic domains. A fundamental rôle is played by the notion of asymmetric event structures that generalize Winskel’s prime event structures, following an idea similar to that of “possible flow ” introduced by Pinna and Poignè. Asymmetric event structures have the usual causal relation of traditional prime event structures, but replace the symmetric conflict with a relation modelling asymmetric conflict or weak causality. Such relation allows one to represent the new kind of dependency between events arising in contextual nets, as well as the usual symmetric conflict. Moreover it is used in a non-trivial way in the definition of the ordering of configurations, which is different from the standard set-inclusion.

We present an event structure semantics for contextual nets, an extension of P/T Petri nets where transitions can check for the presence of tokens without consuming them (read-only operations). A basic rôle is played by asymmetric event structures, a generalization of Winskel’s prime event structures where symmetric conflict is replaced by a relation modelling asymmetric conflict or weak causality, used to represent a new kind of dependency between events arising in contextual nets. Extending Winskel’s seminal work on safe nets, the truly concurrent event based semantics of contextual nets is given at categorical level via a chain of coreflections

In order to model the behaviour of open concurrent systems by means of Petri nets, we introduce open Petri nets, a generalization of the ordinary model where some places, designated as open, represent an interface of the system towards the environment. Besides generalizing the token game to reflect this extension, we define a truly concurrent semantics for open nets by extending the Goltz-Reisig process semantics of Petri nets. We introduce a composition operation over open nets, characterized as a pushout in the corresponding category, suitable to model both interaction through open places and synchronization of transitions. The process semantics is shown to be compositional with respect to such composition operation. Technically, our result is similar to the amalgamation theorem for data-types in the framework of algebraic specifications. A possible application field of the proposed constructions and results is the modeling of interorganizational workflows, recently studied in the literature. This is illustrated by a running example.

In order to model the behaviour of open concurrent systems by means of Petri nets, we introduce open Petri nets, a generalization of the ordinary model where some places, designated as open, represent an interface of the system towards the environment. Besides generalizing the token game to reflect this extension, we define a truly concurrent semantics for open nets by extending the Goltz-Reisig process semantics of Petri nets. We introduce a composition operation over open nets, characterized as a pushout in the corresponding category, suitable to model both interaction through open places and synchronization of transitions. The deterministic process semantics is shown to be compositional with respect to such composition operation. If a net Z 3 results as the composition of two nets Z 1 and Z 2 , having a common subnet Z 0 , then any two deterministic processes of Z 1 and Z 2 which "agree" on the common part, can be "amalgamated" to produce a deterministic process of Z 3 . Vice versa, any deterministic process of Z 3 can be decomposed into processes of the component nets. The amalgamation and decomposition operations are shown to be inverse to each other, leading to a bijective correspondence between the deterministic processes of Z 3 and pair of deterministic processes of Z 1 and Z 2 which agree on the common subnet Z 0 . Technically, our result is similar to the amalgamation theorem for data-types in the framework of algebraic specification. A possible application field of the proposed constructions and results is the modeling of interorganizational workflows, recently studied in the literature. This is illustrated by a running example.

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 doublepushout approach to graph rewriting, namely the derivation trace and the graph process semantics. The notion of concatenable graph process is

. The aim of this paper is an extension of the theory of graph transformation systems in order to make them suitable for the specification of reactive systems. For this purpose two main extensions of the algebraic theory of graph transformations are proposed. Firstly, graph transitions are introduced as a loose interpretation of graph productions, defined using a double pullback construction in contrast to classical graph derivations based on double-pushouts. Two characterisation results relate graph transitions to the classical double-pushout derivations and to amalgamated derivations, respectively. Secondly, a loose semantics for graph transformation systems is defined, which associates with each system a category of models (deterministic transition systems) defined as coalgebras over a suitable functor. Such category has a final object, which includes all finite and infinite transition sequences. The coalgebraic framework makes it possible to introduce a general notion of a logic o...

Abstract. Despite the increasing relevance of model transformation techniques in model-driven software development, research is mainly conducted to the specification and the automation of such transformations. However, since the transformations themselves may also contain conceptual flaws, it is essential to formally analyze them prior to executing them on user models. In the current paper, we focus on a central validation problem of trusted model transformations, namely, termination and propose a Petri net based analysis method that provides a sufficient criterion for the termination problem of model transformations captured by graph transformation systems.