Model Transformation has become central to most software engineering activities. It refers to the process of modifying a (usually graphical) model for the purpose of analysis (by its transformation to some other domain), optimization, evolution, migration or even code generation. In this work, we show termination criteria for model transformation based on graph transformation. This framework offers visual and formal techniques based on rules, in such a way that model transformations can be subject to analysis. Previous results on graph transformation are extended by proving the termination of a transformation if the rules applied meet certain criteria. We show the suitability of the approach by an example in which we translate a simplified version of Statecharts into Petri nets for functional correctness analysis.

The paper presents a case study on the synthesis of labelled transition systems (ltss) for process calculi, choosing as testbed Milner’s Calculus of Communicating System (ccs). The proposal is based on a graphical encoding: each ccs process is mapped into a graph equipped with suitable interfaces, such that the denotation is fully abstract with respect to the usual structural congruence. Graphs with interfaces are amenable to the synthesis mechanism based on borrowed contexts (bcs), proposed by Ehrig and König (which are an instance of relative pushouts, originally introduced by Milner and Leifer). The bc mechanism allows the effective construction of an lts that has graphs with interfaces as both states and labels, and such that the associated bisimilarity is automatically a congruence. Our paper focuses on the analysis of the lts distilled by exploiting the encoding of ccs processes: besides offering some technical contributions towards the simplification of the bc mechanism, the key result of our work is the proof that the bisimilarity on processes obtained via bcs coincides with the standard strong bisimilarity for ccs.

Abstract. Rewriting systems over adhesive categories have been recently introduced as a general framework which encompasses several rewriting-based computational formalisms, including various modelling frameworks for concurrent and distributed systems. Here we begin the development of a truly concurrent semantics for adhesive rewriting systems by defining the fundamental notion of process, well-known from Petri nets and graph grammars. The main result of the paper shows that processes capture the notion of true concurrency—there is a one-toone correspondence between concurrent derivations, where the sequential order of independent steps is immaterial, and (isomorphism classes of) processes. We see this contribution as a step towards a general theory of true concurrency which specialises to the various concrete constructions found in the literature. 1

We introduce a comprehensive operational semantic theory of graph-rewriting. Graph-rewriting here is

Abstract. Adhesive high-level replacement (HLR) system have been recently introduced as a new categorical framework for graph transformation in the double pushout approach [1, 2]. They combine the well-known framework of HLR systems with the framework of adhesive categories introduced by Lack and Sobociński [3, 4]. The main concept behind adhesive categories are the so-called van Kampen squares, which ensure that pushouts along monomorphisms are stable under pullbacks and, vice versa, that pullbacks are stable under combined pushouts and pullbacks. In the case of adhesive HLR categories the class of all monomorphisms is replaced by a subclassMof monomorphisms closed under composition and decomposition. Algebraic high-level nets combine algebraic specifications with Petri nets [5] to allow the modelling of data, data flow and data changes within the net. While elementary Petri nets are adhesive HLR categories, the categories of ordinary place/transition nets and of algebraic high-level nets with fixed signature and algebra only satisfy a weaker version of adhesive HLR categories [6] called weak adhesive HLR categories. The reason is that the category PTNets of place/transition nets has general pullbacks, but pullbacks in general cannot be constructed componentwise in Sets. However, pullbacks along monomorphisms in PTNets can be constructed componentwise in Sets. This is the key idea to weaken the concept of adhesive HLR categories using weak VK squares. In this case, van Kampen squares ensure the corresponding properties only under stricter requirements on the morphisms. Nevertheless, the framework of weak adhesive HLR categories is still sufficient to show under some additional assumptions (which are necessary also in the non-weak case) the following main results:

Abstract. Adhesive categories provide an abstract setting for the doublepushout approach to rewriting, generalising classical approaches to graph transformation. Fundamental results about parallelism and confluence, including the local Church-Rosser theorem, can be proven in adhesive categories, provided that one restricts to linear rules. We identify a class of categories, including most adhesive categories used in rewriting, where those same results can be proven in the presence of rules that are merely left-linear, i.e., rules which can merge different parts of a rewritten object. Such rules naturally emerge, e.g., when using graphical encodings for modelling the operational semantics of process calculi.

In adhesive categories, pushouts along monomorphisms are Van Kampen (vk) squares, a special case of a more general notion called vk-diagram. Other examples of vk-diagrams include coproducts in extensive categories and strict initial objects. Extensive and adhesive categories characterise useful exactness properties of, respectively, coproducts and pushouts along monos and have found several applications in theoretical computer science. We show that the property of being vk is actually universal, not in C but in the bicategory of spans Span C. This theorem of pure category theory sheds light on the nature of spans and suggests promising generalisations of the theory of adhesive categories.

Abstract. Rewriting systems over adhesive categories have been recently introduced as a general framework which encompasses several rewriting-based computational formalisms, including various modelling frameworks for concurrent and distributed systems. Here we begin the development of a truly concurrent semantics for adhesive rewriting systems by defining the fundamental notion of process, well-known from Petri nets and graph grammars. The main result of the paper shows that processes capture the notion of true concurrency—there is a one-toone correspondence between concurrent derivations, where the sequential order of independent steps is immaterial, and (isomorphism classes of) processes. We see this contribution as an important step towards a general theory of true concurrency which specialises to the various concrete constructions found in the literature. 1

The exactness properties of coproducts in extensive categories and pushouts along monos in adhesive categories have found various applications in theoretical computer science, e.g. in program semantics, data type theory and rewriting. We show that these properties can be understood as a single universal property in the associated bicategory of spans. To this end, we first provide a general notion of Van Kampen cocone that specialises to the above colimits. The main result states that Van Kampen cocones can be characterised as exactly those diagrams in C that induce bicolimit diagrams in the bicategory of spans Span C, provided that C has pullbacks and enough colimits.

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