Adhesive High-level Replacement Research Articles

-adhesive transformation systems with nested application conditions. Part 1: parallelism, concurrency and amalgamation

Nested application conditions generalise the well-known negative application conditions and are important for several application domains. In this paper, we present Local Church–Rosser, Parallelism, Concurrency and Amalgamation Theorems for rules with nested application conditions in the framework of$\mathcal{M}$-adhesive categories, where$\mathcal{M}$-adhesive categories are slightly more general than weak adhesive high-level replacement categories. Most of the proofs are based on the corresponding statements for rules without application conditions and two shift lemmas stating that nested application conditions can be shifted over morphisms and rules.

Finitary -adhesive categories

Finitary$\mathcal{M}$-adhesive categories are$\mathcal{M}$-adhesive categories with finite objects only, where$\mathcal{M}$-adhesive categories are a slight generalisation of weak adhesive high-level replacement (HLR) categories. We say an object is finite if it has a finite number of$\mathcal{M}$-subobjects. In this paper, we show that in finitary$\mathcal{M}$-adhesive categories we not only have all the well-known HLR properties of weak adhesive HLR categories, which are already valid for$\mathcal{M}$-adhesive categories, but also all the additional HLR requirements needed to prove classical results including the Local Church-Rosser, Parallelism, Concurrency, Embedding, Extension and Local Confluence Theorems, where the last of these is based on critical pairs. More precisely, we are able to show that finitary$\mathcal{M}$-adhesive categories have a unique$\mathcal{E}$-$\mathcal{M}$factorisation and initial pushouts, and the existence of an$\mathcal{M}$-initial object implies we also have finite coproducts and a unique$\mathcal{E}$′-$\mathcal{M}$pair factorisation. Moreover, we can show that the finitary restriction of each$\mathcal{M}$-adhesive category is a finitary$\mathcal{M}$-adhesive category, and finitarity is preserved under functor and comma category constructions based on$\mathcal{M}$-adhesive categories. This means that all the classical results are also valid for corresponding finitary$\mathcal{M}$-adhesive transformation systems including several kinds of finitary graph and Petri net transformation systems. Finally, we discuss how some of the results can be extended to non-$\mathcal{M}$-adhesive categories.

Parallelism and Concurrency in Adhesive High-Level Replacement Systems with Negative Application Conditions

The goal of this paper is the generalization of parallelism and concurrency results for adhesive High-Level Replacement (HLR) systems to adhesive HLR systems with negative application conditions. These conditions restrict the application of a rule by expressing that a specific structure should not be present before or after applying the rule to a certain context. Such a condition influences thus each rule application or transformation and therefore changes significantly the properties of the replacement system. The effect of negative application conditions on parallelism and concurrency in the replacement system is described in the generalization of the following results, formulated already for adhesive HLR systems without negative application conditions: Local Church-Rosser Theorem, Parallelism Theorem and Concurrency Theorem. These important generalized results will support the development of formal analysis techniques for adhesive HLR systems with negative application conditions.

Towards Algebraic High-Level Systems as Weak Adhesive HLR Categories

Adhesive high-level replacement (HLR) systems have been recently established as a suitable categorical framework for double pushout transformations based on weak adhesive HLR categories. Among different types of graphs and graph-like structures, various kinds of Petri nets and algebraic high-level (AHL) nets are interesting instantiations of adhesive HLR systems. AHL nets combine algebraic specifications with Petri nets to allow the modeling of data, data flow and data changes within the net.For the development and analysis of reconfigurable systems, not only AHL schemas based on an algebraic specification and AHL nets using an additional algebra should be considered, but also AHL systems which additionally include markings of nets.In this paper, we summarize the results for different kinds of AHL schemas and nets, and extend these results to AHL systems. The category of markings is introduced, which allows a general construction combining AHL nets with possible markings leading under certain properties to a weak adhesive HLR category.

Integration of Categorical Frameworks: Rule-Based Refinement and Hierarchical Composition for Components

The integration of two important categorical frameworks – namely adhesive High-Level Replacement (HLR) systems and the generic component concept– yields a categorical approach to component transformation and refinement. The generic component concept is shown to be an adhesive HLR category, so rules and transformations as well as the corresponding results are available. Moreover, the compatibility with the hierarchical component composition is provided. The extension to rule-based refinement requires additional property-preserving morphisms and yields property-preserving rules and transformations, i.e. refinements where compatibility with the hierarchical component composition again is achieved. The categorical framework is instantiated to typed algebraic high-level (AHL) nets and illustrated with an example of AHL net components.

Construction and Properties of Adhesive and Weak Adhesive High-Level Replacement Categories

As presented in Ehrig et al. (Fundamentals of Algebraic Graph Trans- formation, EATCS Monographs, Springer, 2006), adhesive high-level replacement (HLR) categories and systems are an adequate framework for several kinds of trans- formation systems based on the double pushout approach. Since (weak) adhesive HLR categories are closed under product, slice, coslice, comma and functor category constructions, it is possible to build new (weak) adhesive HLR categories from existing ones. But for the general results of transformation systems, as additional properties initial pushouts, binary coproducts compatible with a special morphism class M and a pair factorization are needed to obtain the full theory. In this paper, we analyze under which conditions these additional properties are preserved by the categorical constructions in order to avoid checking these properties explicitly.

On Term Graphs as an Adhesive Category

The recent interest in bisimulation congruences for reduction systems, stimulated by the research on general (often graphical) frameworks for nominal calculi, has brought forward many proposals for categorical formalisms where relevant properties of observational equivalences could be auto- matically verified.Interestingly, some of these formalisms also identified suitable categories where the standard tools and techniques developed for the double-pushout approach to graph transformation [A. Corradini, U. Montanari, F. Rossi, H. Ehrig, R. Heckel and M. Löwe, Algebraic approaches to graph transformation I: Basic concepts and double pushout approach, in: G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformation, I: Foundations, World Scientific, 1997, pp. 163–246] could be recast, thus providing a valid alternative to the High-Level Replacement Systems paradigm [H. Ehrig, A. Habel, H.-J. Kreowski and F. Parisi-Presicce, Parallelism and concurrency in highlevel replacement systems, Mathematical Structures in Computer Science 1 (1991) 361–404].In this paper we consider the category of term graphs, and we prove that it (partly) fits in the general framework for adhesive categories, developed in [S. Lack, and P. Sobociński, Adhesive categories, in: I. Walukiewicz, editor, Foundations of Software Science and Computation Structures, Lect. Notes in Comp. Sci. 2987 (2004), pp. 273–288, P. Sobociński, “Deriving bisimulation congruences from reduction systems”, Ph.D. thesis, BRICS, Department of Computer Science, University of Aaurhus (2004)], extended in [H. Ehrig, A. Habel, J. Padberg and U. Prange, Adhesive high-level replacement categories and systems, in: G. Engels and F. Parisi-Presicce, editors, Graph Transformation, Lect. Notes in Comp. Sci. (2004)] and applied to reduction systems in [V. Sassone, and P. Sobociński, Congruences for contextual graph-rewriting, Technical Report RS-04-11, BRICS, Department of Computer Science, University of Aarhus (2004)]. The main technical achievement concerns the proof that the category of term graphs is actually quasi-adhesive, obtained by proving the existence of suitable Van Kampen squares.