In the area of Petri nets, many different developments have taken place within the last 30 years, in academia as well as in practice. For an adequate use in practice, a coherent and application oriented combination of various types and techniques for Petri nets is necessary. In order to attain a formal basis for different classes of Petri nets we introduce the concept of abstract Petri nets. The essential point of abstract Petri nets is to allow different kinds of net structures as well as the combination of various kinds of data types. This means that in abstract Petri nets the data type and the net structure part can be considered as abstract parameters which can be instantiated to different concrete net classes. We show that several net classes, like place/transition nets, elementary nets, S-graphs, algebraic high-level net...

this paper we concentrate on structuring and refinement concepts for graph transformation systems. Conceptually, we distinguish between two kinds of structuring. We speak of horizontal structuring if a large specification is obtained by combining and modifying smaller ones, possibly sharing some common parts. Instead, if we consider the relationship between a more abstract and a more concrete version of the same specification, or between a specification and its implementation, we speak of vertical structuring.

Adhesive high-level replacement (HLR) categories and systems are introduced as a new categorical framework for graph transformation in a broad sense, which combines the well-known concept of HLR systems with the new concept of adhesive categories introduced by Lack and Sobociński. In this paper we show that most of the HLR properties, which had been introduced ad hoc to generalize some basic results from the category of graphs to high-level structures, are valid already in adhesive HLR categories. As a main new result in a categorical framework we show the Critical Pair Lemma for local confluence of transformations. Moreover we present a new version of embeddings and extensions for transformations in our framework of adhesive HLR systems.

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.

The concept of rule-based modification developed in the area of algebraic graph transformations and high-level replacement systems has recently shown to be a powerful concept for vertical stucturing of Petri nets. This includes low-level and high-level Petri nets, especially algebraic high-level nets which can be considered as an integration of algebraic specifications and Petri nets. In a large case study rule-based modi cation of algebraic high-level nets has been applied successfully for the requirements analysis of a medical information system. The main new result in this paper extends rule-based modification of algebraic high-level nets such that it preserves safety properties formulated in terms of temporal logic. For software development based on rule-based modi cation of algebraic high-level nets as a vertical development strategy this extension is an important new technique. It is called rule-based re nement. As a running example an important safety property of a medical information system is considered and is shown to be preserved under rule-based refinement.

The new concept of parameterized net classes is introduced in order to allow a uniform presentation of different kinds of Petri net classes. By different actualizations of the net structure parameter and the data type parameter we obtain several wellknown net classes, like elementary nets, place-transition nets, coloured Petri nets, predicate /transition nets, and algebraic high-level nets, as well as several interesting new classes of low- and high-level nets. While the concept of parameterized net classes is defined on a purely set theoretical level, we also sketch an extended concept of universal parameterized net classes taking into account also morphisms and universal properties in the sense of category theory. The extended concept, presented in a sperate paper, leads to a uniform theory of constructions and compatibility results concerning union and fusion of nets for different types of net classes.

This paper is concerned with the application of a formal technique to software engineering. In this case study we have used rule-based refinement of algebraic high-level nets for the requirements engineering of a medical information system. We outline the basic ideas of rule-based refinement and discuss how this technique is applied to the development from actual state analysis to functional essence. INTRODUCTION The main idea of requirements engineering in general is the analysis of the system leading to a documentation of the actual state and a transformation of the actual state to the functional essence of the corresponding system (see [DeM78, Dav90, Huß93]). Hence, requirements engineering is a stepwise abstraction of the actual state leading to the functional essence. In this paper we start with a documentation of the actual state of an information system by high-level Petri nets [JR91, Jen95], and more precisely by algebraic high-level nets in the spirit of [Vau86, KS91, Rei91, ...

Based on the well-known theory of high-level replacement systems -- a categorical formulation of graph grammars -- we present new results concerning refinement of high-level replacement systems. Motivated by Petri nets, where refinement is often given by morphisms, we give a categorical notion of refinement. This concept is called Q-transformations and is established within the framework of high-level replacement systems. The main idea is to supply rules with an additional morphism, which belongs to a specific class Q of morphisms. This leads to the new notions of Q-rules and Q-transformations. Moreover, several concepts and results of high-level replacement systems are extended to Q-transformations. These are sequential and parallel transformations, union, and fusion, based on different colimit constructions. The main results concern the compatibility of these constructions with Q-transformations that is the corresponding theorems for usual transformations are extended to Q-transform...

In this paper we present an algebraic approach to statecharts as they are used in the Statemate tool in the style of "Petri-Nets are Monoids" for place-transition nets developed by Meseguer and Montanari. We apply the framework of high-level-replacement systems, a categorical generalization of graph transformation systems, in order to de#ne union as horizontal as well as transformation and re#nementasvertical structuring techniques for statecharts. The #rst main result shows compatibility of union and transformation in a suitable category of statecharts. We present an algorithm for the computation of all transitions enabled within one step. The second main result shows the correctness of this algorithm. We de#ne re#nement morphisms for statecharts, which allow re#nement of arbitrary states, in contrast to concepts in the literature where only basic and root states are subject of re#nement. The third main result shows that re#nement morphisms are compatible with the behavio...