High-Level Replacement Systems Applied to Algebraic Specifications and Petri Nets

Abstract

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.

Deriving Bisimulation Congruences in the DPO Approach to Graph Rewriting

“…Our results do not only hold in graph structure categories, but also in other categories which satisfy certain properties typical for the categories of sets and high-level replacement systems [4]. In this context it is also interesting to point out that most of the categorical properties we need hold already in adhesive categories [12].…”

Adhesive and quasiadhesive categories

“…Indeed, they are usually assumed as axioms (see [9] and §8 below) in attempts at generalizing graph rewriting. They hold in any adhesive category by Lemma 2.3.…”

“…Once such a diagram is constructed we may deduce that C D, that is, C rewrites to D. DPO rewriting is formulated in categorical terms and is therefore portable to structures other than directed graphs. There have been several attempts [9,11] to isolate classes of categories in which one can perform DPO rewriting and in which one can show that such rewriting grammars satisfy useful properties. In particular, several axioms were put forward in [11] in order to prove a local Church-Rosser theorem for such general grammars.…”

“…Sometimes referred to as generalized union [9], it can often be thought of as the construction of a larger structure from two smaller structures by gluing them together along a shared substructure.…”

“…We prove the local Church-Rosser theorem and the concurrency theorem without the need for extra axioms. We shall also examine how adhesive and quasiadhesive categories fit within the previously conceived general frameworks for rewriting [9,11]. Many of the axioms put forward in [11] follow elegantly as lemmas from the axioms of adhesive categories.…”