Using High-Level Replacement Systems to Preserve Safety Properties in PlaceTransition Net T

1 Introduction
The generalization of graph transformations, namely high-level replacement (HLR) systems can successfully be employed in other areas, as Petri nets, to obtain results that are relevant for software engineering. In this paper we apply an extension of high-level replacement systems to Petri nets in order to achieve an integration of transformations, and hence all its advantages, with the preservation of (safety) properties. In contrast to the graph transformation approach, where rules and transformations describe dynamic behavior, in the area of Petri nets we use rules and transformations to represent stepwise modi cation of nets. This is considered to be a vertical structuring technique in software engineering and is known as rule-based modi cation. But note, rule-based modi cation does not allow preserving system properties, and thus cannot be considered a re nement technique. The combination of transformations with the preservation of safety properties is very relevant in software engineering as the veri cation of large and complex systems is often necessary, but very complicated (if possible at all) and thus expensive. Thus, it is desirable to state and prove (safety) properties at an early state. The idea of developing a system while preserving safety properties is illustrated in section 2, in which we introduce a small example. We give a place/transition net that models a very simple elevator and satis es safety properties concerning the opening and closing of the doors to the elevator shaft. Then we stepwise introduce exclusive requests for opening the doors using rules and transformations that preserve safety properties. Thus we gain a model of the elevator with request, that satis es the safety properties concerning the doors without proving these properties again for this new model. High-level replacement systems have been introduced in EHKP91] as a categorical generalization of the double pushout approach to graph transformations, there the rst application to place/transition nets has been given as well. The application of high-level replacement systems to a special domain as place/transition nets, algebraic speci cation etc. requires a suitable category and the satisfaction of the HLR-conditions. In the case of place/transition nets with an initial marking this follows as a direct consequence of the results presented in RP94]. In Pad96] high-level replacement systems have been extended in order to allow di erent kinds of morphisms within the transformation step. This enables the expression of re nement using a morphism, called Q-morphism, directly from the left hand side of the rule to the right hand side. Based on this extension we can integrate net transformations with place preserving morphisms Peu97], that preserve safety properties in terms of speci c temporal logic formulas. The main result of this paper is the integration of the place preserving morphisms with the rules and transformations. This is achieved by proving three nontrivial conditions, called Q-conditions, see fact 4.6, fact 4.7 and fact 4.8. These proofs require detailed information about the involved categories, morphisms
Using High-Level Replacement Systems to Preserve Safety Properties in Place/Transition Net Transformations
e-mail: fpadberg, gajewskyg@cs.tu-berlin.de
J. Padberg, M. Gajewsky1 Technical University of Berlin
and constructions, and are { due to space limitations { omitted in this paper. These technical result yield the important outcome for software engineering, stated in thm. 4.12 (Q-Transformations Preserve Safety Properties), that the concept of rules together with place preserving morphisms and transformations, allows the preservation of safety properties in the sense of MP92] given as a temporal logic formula. The paper is organize 2 illustrates the main ideas in terms of a small example. Then we review high-level replacement systems in section 3. Subsequently we give place/transition nets together with morphisms in section 4 and then we show that place/transition nets together with place preserving morphisms satisfy the Q-conditions, leading to the main theorem. As a conclusion we discuss future work in section 5.
1 This work is part of the joint research project \DFG-Forschergruppe Petrinetz-Technologie" between H. Weber (Coordinator), H. Ehrig (both from the Technical University Berlin) and W. Reisig (Humboldt-Universitat zu Berlin), supported by the German Research Council (DFG).