The general idea of high-level replacement systems is to generalize the concept of graph transformation systems and graph grammars from graphs to all kinds of structures which are of interest in Computer Science and Mathematics. Within the algebraic approach of graph transformation this is possible by replacing graphs, graph morphisms, and pushouts (gluing) of graphs by objects, morphisms, and pushouts in a suitable category. Of special interest are categories for all kinds of labelled and typed graphs, hypergraphs, algebraic speci cations and Petri nets. In this chapter, we show how some basic results for graph transformation systems in the algebraic double pushout approach can be reformulated in the framework of high-level replacement systems. The speci c choice of results concerning local Church-Rosser properties and horizontal structuring is motivated by the results needed in our application areas studied in this contribution. In order to show the great variety of the high-level replacement approach we do not consider speci c graphs and graph transformation but algebraic speci cations and Petri nets as application domains, where transformation corresponds to rule-based changes of the structure of speci cations and nets, respectively. The rst application shows how high-level replacement systems can be instantiated by algebraic speci cations. An algebraic transformation rule corresponds to the interface part of an algebraic module speci cation for software systems. This allows applying high-level replacement techniques to software system design. As an application it is shown how to reuse an algebraic module speci cation of an airport schedule for the design of a 341 342 CHAPTER 6. HIGH-LEVEL REPLACEMENT SYSTEMS : : : book library. The second main application shows how rule-based modi cation of Petri nets can be considered as a special case of high-level replacement techniques. An important result is the compatibility of horizontal structuring of nets with rule-based modi cation. This result is essential within a case study of a medical information system where the functional essence is developed by rule-based modi cation from the actual state of the system represented by algebraic high-level nets.

Abstract. 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 modification 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 highlevel nets such that it preserves safety properties formulated in terms of temporal logic. For software development based on rule-based modification of algebraic high-level nets as a vertical development strategy this extension is an important new technique. It is called rule-based refinement. 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.

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Our approach of rule-based refinement 1 provides a formal description for the stepwise system development based on Petri nets. Rules with a left-hand and a right-hand side allow replacing subnets in a given algebraic high-level net system. The extension of these rules supports the preservation of system properties. In this paper we extend the preservation of safety properties significantly. We define rules, that introduce new safety properties. In our new approach we propose first the verification of properties at the moment they can be expressed and then their preservation further on. Hence, properties can be checked as long as the system is still small. Moreover, introducing properties allows checking these for the relevant subpart only. Changes that are required later on can be treated the same way and hence preserve the system properties. Hence, we have made a step towards a formal technique for the stepwise system development during analysis and design.

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